Tam giác Pascal là một trong những ví dụ kinh điển được dạy cho sinh viên kỹ thuật. Nó có nhiều cách hiểu. Một trong những cách nổi tiếng là việc sử dụng nó với các phương trình nhị thức
Tất cả các giá trị bên ngoài tam giác được coi là không [0]. Hàng đầu tiên là 0 1 0 trong khi chỉ có 1 có khoảng trắng trong tam giác pascal, 0 là vô hình. Hàng thứ hai có được bằng cách cộng [0+1] và [1+0]. Đầu ra được kẹp giữa hai số không. Quá trình tiếp tục cho đến khi đạt được mức yêu cầu
Tam giác Pascal có thể được suy ra bằng định lý nhị thức. Chúng ta có thể sử dụng các kết hợp và giai thừa để đạt được điều này
thuật toán
Giả sử rằng chúng ta đã biết rõ về giai thừa, chúng ta sẽ xem xét khái niệm cốt lõi của việc vẽ một tam giác pascal theo kiểu từng bước -
START Step 1 - Take number of rows to be printed, n. Step 2 - Make outer iteration I for n times to print rows Step 3 - Make inner iteration for J to [N - 1] Step 4 - Print single blank space " " Step 5 - Close inner loop Step 6 - Make inner iteration for J to I Step 7 - Print nCr of I and J Step 8 - Close inner loop Step 9 - Print NEWLINE character after each inner iteration Step 10 - Return STOP
mã giả
Chúng ta có thể rút ra một mã giả cho thuật toán đã đề cập ở trên, như sau -
procedure pascals_triangle FOR I = 0 to N DO FOR J = 0 to N-1 DO PRINT " " END FOR FOR J = 0 to I DO PRINT nCr[i,j] END FOR PRINT NEWLINE END FOR end procedure
Thực hiện
Hãy thực hiện toàn bộ chương trình này. Chúng ta sẽ triển khai các hàm cho giai thừa [không đệ quy] cũng như ncr [kết hợp]
Vòng lặp bên trong đầu tiên hiển thị không gian trên màn hình đầu ra. nếu bạn không muốn hiển thị đầu ra ở giữa màn hình thì hãy xóa cái này để lặp lại
Trong tam giác pascal, ở mỗi hàng, số đầu tiên và số cuối cùng là 1 và các số còn lại là tổng của hai số ngay phía trên nó. Vì vậy, bên trong vòng lặp bên trong thứ hai, chúng tôi viết điều kiện dưới đây,
if[j==0||j==i] pascal[i][j]=1;
else pascal[i][j] = pascal[i-1][j-1] + pascal[i-1][j];
Lưu ý rằng trong mỗi hàng, kích thước của mảng là n, nhưng ở hàng đầu tiên, phần tử đầu tiên duy nhất được lấp đầy và phần còn lại có giá trị rác. Tương tự, ở hàng thứ hai, chỉ phần tử thứ nhất và thứ hai của mảng được lấp đầy và phần còn lại có giá trị rác. Chúng tôi không muốn hiển thị giá trị rác. Vì vậy, chúng ta sẽ dễ dàng hiển thị đầu ra tại thời điểm tính toán
Tìm hiểu thêm. - Những bí mật toán học của tam giác Pascal
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Tam giác Pascal là một mảng tam giác các hệ số nhị thức. Viết hàm nhập vào một giá trị nguyên n và in ra n dòng đầu tiên của tam giác Pascal. Sau đây là 6 hàng đầu tiên của Tam giác Pascal
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1
Đề nghị thực hành
Tam giác Pascal
Thử nó
Phương pháp 1 [ Độ phức tạp thời gian O[n^3] ]
Số lượng mục trong mỗi dòng bằng số dòng. Ví dụ, dòng đầu tiên có “1”, dòng thứ hai có “1 1”, dòng thứ ba có “1 2 1”,. và như thế. Mỗi mục trong một dòng là giá trị của Hệ số nhị thức. Giá trị của mục thứ i trong dòng số dòng là C[line, i]. Giá trị có thể được tính bằng công thức sau.
C[line, i] = line! / [ [line-i]! * i! ]
Một phương pháp đơn giản là chạy hai vòng lặp và tính giá trị của Hệ số nhị thức trong vòng lặp bên trong.
C++
// C++ code for Pascal's Triangle
#include
using
namespace
C[line, i] = line! / [ [line-i]! * i! ]0
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]2
C[line, i] = line! / [ [line-i]! * i! ]3
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]5_______4_______4
C[line, i] = line! / [ [line-i]! * i! ]7
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]9
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 11
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 12
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 13
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15_______4_______4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 10
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 12
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time1
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time3
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time5
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 10
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
1 1 1 1 2 1 1 3 3 1 1 4 6 4 12
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 12
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
// C++ code for Pascal's Triangle
3
C[line, i] = line! / [ [line-i]! * i! ]3
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]5_______4_______4
C[line, i] = line! / [ [line-i]! * i! ]7
C[line, i] = line! / [ [line-i]! * i! ]4
#include
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
#include
41 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
6 #include
71 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
91 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
using
41 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
8C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
namespace
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
namespace
5 namespace
6// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
namespace
9
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]01
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]05
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]07
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
namespace
5 C[line, i] = line! / [ [line-i]! * i! ]10
// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]13
C
// C++ code for Pascal's Triangle
C[line, i] = line! / [ [line-i]! * i! ]15
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]2
C[line, i] = line! / [ [line-i]! * i! ]3
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]5_______4_______4
C[line, i] = line! / [ [line-i]! * i! ]7
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]9
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 11
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 12
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 13
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15_______4_______4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 10
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 12
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time1
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time3
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time5
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 10
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
C[line, i] = line! / [ [line-i]! * i! ]57
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]59
C[line, i] = line! / [ [line-i]! * i! ]60
C[line, i] = line! / [ [line-i]! * i! ]61_______4_______62
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]57
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
C[line, i] = line! / [ [line-i]! * i! ]67
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
// C++ code for Pascal's Triangle
3
C[line, i] = line! / [ [line-i]! * i! ]3
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]5_______4_______4
C[line, i] = line! / [ [line-i]! * i! ]7
C[line, i] = line! / [ [line-i]! * i! ]4
#include
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
#include
41 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
6 #include
71 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
91 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
using
41 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
8C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
namespace
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
namespace
5 namespace
6// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
namespace
9
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]01
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]05
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]07
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
namespace
5 C[line, i] = line! / [ [line-i]! * i! ]10
// C++ code for Pascal's Triangle
0
Java
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 121
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 122
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 123
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 125
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 126
namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 11
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 142
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 144
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 149
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 151
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time1
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 157
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 162
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 164
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 166
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 167
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 168
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
C[line, i] = line! / [ [line-i]! * i! ]67
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 171
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 173
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 180
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 182
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]5_______4_______4
C[line, i] = line! / [ [line-i]! * i! ]7
C[line, i] = line! / [ [line-i]! * i! ]4
#include
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 195
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
#include
6 #include
7C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
#include
91 1 1 1 2 1 1 3 3 1 1 4 6 4 104
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 162
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
1 1 1 1 2 1 1 3 3 1 1 4 6 4 111
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
using
81 1 1 1 2 1 1 3 3 1 1 4 6 4 11
1 1 1 1 2 1 1 3 3 1 1 4 6 4 117
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ]67
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
// C++ code for Pascal's Triangle
0C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
namespace
5 namespace
61 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 129
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 134
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 139
1 1 1 1 2 1 1 3 3 1 1 4 6 4 140
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]07
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 148
Python3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 149
1 1 1 1 2 1 1 3 3 1 1 4 6 4 150
1 1 1 1 2 1 1 3 3 1 1 4 6 4 151
1 1 1 1 2 1 1 3 3 1 1 4 6 4 152
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 154
1 1 1 1 2 1 1 3 3 1 1 4 6 4 155
1 1 1 1 2 1 1 3 3 1 1 4 6 4 152
1 1 1 1 2 1 1 3 3 1 1 4 6 4 157
1 1 1 1 2 1 1 3 3 1 1 4 6 4 158
namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 161
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 163
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 166
1 1 1 1 2 1 1 3 3 1 1 4 6 4 167
1 1 1 1 2 1 1 3 3 1 1 4 6 4 168
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
1 1 1 1 2 1 1 3 3 1 1 4 6 4 171
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 174
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 176
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 178
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 181
1 1 1 1 2 1 1 3 3 1 1 4 6 4 167
1 1 1 1 2 1 1 3 3 1 1 4 6 4 168
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
1 1 1 1 2 1 1 3 3 1 1 4 6 4 186
1 1 1 1 2 1 1 3 3 1 1 4 6 4 187
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 189
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11_______25_______91
1 1 1 1 2 1 1 3 3 1 1 4 6 4 192
1 1 1 1 2 1 1 3 3 1 1 4 6 4 193
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
1 1 1 1 2 1 1 3 3 1 1 4 6 4 195
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 197
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 191
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time00
namespace
3
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time03
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time04
1 1 1 1 2 1 1 3 3 1 1 4 6 4 157
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time06
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time08
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
6 ________35 _______13________35 _______14 ________35 ______15C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time17_______25_______96
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time19
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time14
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 181_______25_______67
1 1 1 1 2 1 1 3 3 1 1 4 6 4 168
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time29
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time08
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time08
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time34
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time35
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time14
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time37
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time08
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time08
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time42
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time42
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time44
1 1 1 1 2 1 1 3 3 1 1 4 6 4 187
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time47
namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
namespace
5 C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time08
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time53
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 140
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time57
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time60
C#
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time61
using
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time63
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 125
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 126
namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 11
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 142
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 144
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time1
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 157
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 10
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 102
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 167
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 168
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
C[line, i] = line! / [ [line-i]! * i! ]67
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 171
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 109
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 180
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 182
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]5_______4_______4
C[line, i] = line! / [ [line-i]! * i! ]7
C[line, i] = line! / [ [line-i]! * i! ]4
#include
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]4
#include
41 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
#include
6 #include
7C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
#include
91 1 1 1 2 1 1 3 3 1 1 4 6 4 104
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
using
4C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
using
81 1 1 1 2 1 1 3 3 1 1 4 6 4 11
namespace
0C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
// C++ code for Pascal's Triangle
0C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
namespace
5 namespace
61 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 129
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 164
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]05
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]07
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 176
PHP
1 1 1 1 2 1 1 3 3 1 1 4 6 4 177
1 1 1 1 2 1 1 3 3 1 1 4 6 4 178
1 1 1 1 2 1 1 3 3 1 1 4 6 4 179
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 182
C[line, i] = line! / [ [line-i]! * i! ]5_______25_______84
C[line, i] = line! / [ [line-i]! * i! ]60
1 1 1 1 2 1 1 3 3 1 1 4 6 4 186
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time47
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 190
1 1 1 1 2 1 1 3 3 1 1 4 6 4 191
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
6 1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 186
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 184
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 186
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time47
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 186
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 184
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 186
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
// C++ code for Pascal's Triangle
11 // C++ code for Pascal's Triangle
12// C++ code for Pascal's Triangle
11 // C++ code for Pascal's Triangle
141 1 1 1 2 1 1 3 3 1 1 4 6 4 186
// C++ code for Pascal's Triangle
16// C++ code for Pascal's Triangle
11C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time47
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 190
// C++ code for Pascal's Triangle
231 1 1 1 2 1 1 3 3 1 1 4 6 4 184
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time14
// C++ code for Pascal's Triangle
11C[line, i] = line! / [ [line-i]! * i! ]67
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 190
// C++ code for Pascal's Triangle
30// C++ code for Pascal's Triangle
11 // C++ code for Pascal's Triangle
321 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0namespace
5
1 1 1 1 2 1 1 3 3 1 1 4 6 4 190
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 11
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 12
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 13
1 1 1 1 2 1 1 3 3 1 1 4 6 4 182
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 184
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time47
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 10
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 12
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
// C++ code for Pascal's Triangle
55 // C++ code for Pascal's Triangle
12// C++ code for Pascal's Triangle
55 // C++ code for Pascal's Triangle
141 1 1 1 2 1 1 3 3 1 1 4 6 4 184
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
// C++ code for Pascal's Triangle
55// C++ code for Pascal's Triangle
621 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time1
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time3
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time5
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
// C++ code for Pascal's Triangle
11 // C++ code for Pascal's Triangle
12// C++ code for Pascal's Triangle
11 // C++ code for Pascal's Triangle
77// C++ code for Pascal's Triangle
551 1 1 1 2 1 1 3 3 1 1 4 6 4 18
// C++ code for Pascal's Triangle
11// C++ code for Pascal's Triangle
621 1 1 1 2 1 1 3 3 1 1 4 6 4 193
// C++ code for Pascal's Triangle
83 // C++ code for Pascal's Triangle
84// C++ code for Pascal's Triangle
85// C++ code for Pascal's Triangle
55C[line, i] = line! / [ [line-i]! * i! ]60
// C++ code for Pascal's Triangle
11// C++ code for Pascal's Triangle
891 1 1 1 2 1 1 3 3 1 1 4 6 4 13
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
// C++ code for Pascal's Triangle
92
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
// C++ code for Pascal's Triangle
83 1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
#include
01
1 1 1 1 2 1 1 3 3 1 1 4 6 4 184
#include
031 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 184
C[line, i] = line! / [ [line-i]! * i! ]67
C[line, i] = line! / [ [line-i]! * i! ]1
#include
08
#include
09
Javascript
#include
10
C[line, i] = line! / [ [line-i]! * i! ]1
#include
12
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 11
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 182
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time57
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 142
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 144
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
#include
301 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time1
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 157
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
#include
39C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
#include
411 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 167
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 168
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
C[line, i] = line! / [ [line-i]! * i! ]67
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 171
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
#include
48#include
49_______4_______671 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 180
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 182
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 182
#include
621 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
#include
661 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
#include
6 #include
7C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
#include
91 1 1 1 2 1 1 3 3 1 1 4 6 4 104
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
#include
76C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
using
81 1 1 1 2 1 1 3 3 1 1 4 6 4 11
namespace
0C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
// C++ code for Pascal's Triangle
0C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
namespace
5 namespace
61 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]1
#include
01
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
951 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]07
C[line, i] = line! / [ [line-i]! * i! ]1
#include
99
Đầu ra
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1
Độ phức tạp về thời gian. O[n^3]
Không gian phụ trợ. Ô[1]
Phương pháp 2[ O[n^2] thời gian và O[n^2] không gian thêm]
Nếu chúng ta nhìn kỹ hơn vào tam giác, chúng ta sẽ quan sát thấy rằng mọi mục nhập đều là tổng của hai giá trị phía trên nó. Vì vậy, chúng ta có thể tạo một mảng 2D lưu trữ các giá trị được tạo trước đó. Để tạo một giá trị trong một dòng, chúng ta có thể sử dụng các giá trị được lưu trữ trước đó từ mảng.
C++
using
00
using
01
using
02
using
03
using
namespace
C[line, i] = line! / [ [line-i]! * i! ]0
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15_______4_______4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
151 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
171 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
using
20C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
231 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
251 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
34C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
36C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 10
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
45C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
#include
6 using
481 1 1 1 2 1 1 3 3 1 1 4 6 4 11____1562_______50
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
52C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
54C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
561 1 1 1 2 1 1 3 3 1 1 4 6 4 11
using
58using
59_______1562_______60
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
621 1 1 1 2 1 1 3 3 1 1 4 6 4 13
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
// C++ code for Pascal's Triangle
0C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 12
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 129
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]01
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
using
811 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]07
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
namespace
5 C[line, i] = line! / [ [line-i]! * i! ]10
// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
using
89
C
using
90
using
01
using
02
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15_______4_______4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
using
15
using
17
C[line, i] = line! / [ [line-i]! * i! ]4
using
20C[line, i] = line! / [ [line-i]! * i! ]1
namespace
03
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
341 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
361 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 10
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
451 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
6 using
48C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
501 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
521 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
541 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
56C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
namespace
341 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]57
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]59
namespace
391 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]57
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
C[line, i] = line! / [ [line-i]! * i! ]67
// C++ code for Pascal's Triangle
0
// C++ code for Pascal's Triangle
0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 129
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]01
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ]4
using
811 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]07
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
namespace
5 C[line, i] = line! / [ [line-i]! * i! ]10
// C++ code for Pascal's Triangle
0
Java
namespace
61
namespace
62
namespace
63
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 122
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 123
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 125
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 126
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
namespace
73C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 139
namespace
771 1 1 1 2 1 1 3 3 1 1 4 6 4 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]07
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15_______4_______4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
namespace
91
C[line, i] = line! / [ [line-i]! * i! ]4
namespace
93namespace
94 C[line, i] = line! / [ [line-i]! * i! ]4
namespace
96C[line, i] = line! / [ [line-i]! * i! ]1
namespace
03
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 149
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 151
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]007
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 162
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 164
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
451 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
6 C[line, i] = line! / [ [line-i]! * i! ]021
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time47
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]025
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
56 C[line, i] = line! / [ [line-i]! * i! ]030
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]032
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ]034
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ]036
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ]038
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]040
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]044
// C++ code for Pascal's Triangle
84C[line, i] = line! / [ [line-i]! * i! ]67
// C++ code for Pascal's Triangle
0
// C++ code for Pascal's Triangle
0
// C++ code for Pascal's Triangle
0
Python3
C[line, i] = line! / [ [line-i]! * i! ]050
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]052
C[line, i] = line! / [ [line-i]! * i! ]053
1 1 1 1 2 1 1 3 3 1 1 4 6 4 157
C[line, i] = line! / [ [line-i]! * i! ]055
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]057
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]060
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]062
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]064
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
C[line, i] = line! / [ [line-i]! * i! ]066
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
C[line, i] = line! / [ [line-i]! * i! ]069
1 1 1 1 2 1 1 3 3 1 1 4 6 4 167
1 1 1 1 2 1 1 3 3 1 1 4 6 4 168
C[line, i] = line! / [ [line-i]! * i! ]072
C[line, i] = line! / [ [line-i]! * i! ]073
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
C[line, i] = line! / [ [line-i]! * i! ]075
1 1 1 1 2 1 1 3 3 1 1 4 6 4 167
1 1 1 1 2 1 1 3 3 1 1 4 6 4 168
C[line, i] = line! / [ [line-i]! * i! ]072
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19____4_______081
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19____4_______083
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 166
1 1 1 1 2 1 1 3 3 1 1 4 6 4 167
1 1 1 1 2 1 1 3 3 1 1 4 6 4 168
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
C[line, i] = line! / [ [line-i]! * i! ]091
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 174
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]096
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 181
1 1 1 1 2 1 1 3 3 1 1 4 6 4 167
1 1 1 1 2 1 1 3 3 1 1 4 6 4 168
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
1 1 1 1 2 1 1 3 3 1 1 4 6 4 186
1 1 1 1 2 1 1 3 3 1 1 4 6 4 187
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ]057
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11____4_______110
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11____4_______112
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
#include
6C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time44_______4_______116
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 150
C[line, i] = line! / [ [line-i]! * i! ]118
1 1 1 1 2 1 1 3 3 1 1 4 6 4 181
C[line, i] = line! / [ [line-i]! * i! ]116
C[line, i] = line! / [ [line-i]! * i! ]121
1 1 1 1 2 1 1 3 3 1 1 4 6 4 193_______4_______123____25_______96
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 193
1 1 1 1 2 1 1 3 3 1 1 4 6 4 191
C[line, i] = line! / [ [line-i]! * i! ]128
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
C[line, i] = line! / [ [line-i]! * i! ]131
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11____4_______134
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
C[line, i] = line! / [ [line-i]! * i! ]136
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11____1562_______56
C[line, i] = line! / [ [line-i]! * i! ]139
1 1 1 1 2 1 1 3 3 1 1 4 6 4 193
C[line, i] = line! / [ [line-i]! * i! ]123
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
C[line, i] = line! / [ [line-i]! * i! ]143
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ]146
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ]149
1 1 1 1 2 1 1 3 3 1 1 4 6 4 187
C[line, i] = line! / [ [line-i]! * i! ]151
C[line, i] = line! / [ [line-i]! * i! ]152
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ]155
1 1 1 1 2 1 1 3 3 1 1 4 6 4 193
1 1 1 1 2 1 1 3 3 1 1 4 6 4 191
C[line, i] = line! / [ [line-i]! * i! ]128
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
C[line, i] = line! / [ [line-i]! * i! ]161
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 191
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 195
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 197
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]170
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
namespace
77C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time57
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]176
C[line, i] = line! / [ [line-i]! * i! ]177
C#
C[line, i] = line! / [ [line-i]! * i! ]178
namespace
62
namespace
63
using
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time63
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 125
C[line, i] = line! / [ [line-i]! * i! ]185
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15_______4_______4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
namespace
3
using
15
using
17
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]198
namespace
94 C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]201
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 142
C[line, i] = line! / [ [line-i]! * i! ]204
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 157
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 10
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]223
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]225
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
6 using
48C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]230
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
56 C[line, i] = line! / [ [line-i]! * i! ]233
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140_______4_______235
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]237
C[line, i] = line! / [ [line-i]! * i! ]238______4_______239
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]241
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0C[line, i] = line! / [ [line-i]! * i! ]244____1560_______84
C[line, i] = line! / [ [line-i]! * i! ]67
// C++ code for Pascal's Triangle
0
// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
#include
01
1 1 1 1 2 1 1 3 3 1 1 4 6 4 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
C[line, i] = line! / [ [line-i]! * i! ]254
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
using
811 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]07
// C++ code for Pascal's Triangle
0
// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]264
C[line, i] = line! / [ [line-i]! * i! ]265
PHP
1 1 1 1 2 1 1 3 3 1 1 4 6 4 177
C[line, i] = line! / [ [line-i]! * i! ]267
using
01
using
02
1 1 1 1 2 1 1 3 3 1 1 4 6 4 182
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 184
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time47
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
151 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
171 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]280
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
C[line, i] = line! / [ [line-i]! * i! ]282
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]282
C[line, i] = line! / [ [line-i]! * i! ]285
namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
231 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
251 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
// C++ code for Pascal's Triangle
55 // C++ code for Pascal's Triangle
12// C++ code for Pascal's Triangle
55 // C++ code for Pascal's Triangle
141 1 1 1 2 1 1 3 3 1 1 4 6 4 184
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
// C++ code for Pascal's Triangle
55// C++ code for Pascal's Triangle
621 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
34C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
36C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
// C++ code for Pascal's Triangle
11 // C++ code for Pascal's Triangle
12// C++ code for Pascal's Triangle
11 // C++ code for Pascal's Triangle
77// C++ code for Pascal's Triangle
551 1 1 1 2 1 1 3 3 1 1 4 6 4 18
// C++ code for Pascal's Triangle
11// C++ code for Pascal's Triangle
62C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
using
451 1 1 1 2 1 1 3 3 1 1 4 6 4 11
#include
6 1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
// C++ code for Pascal's Triangle
55 C[line, i] = line! / [ [line-i]! * i! ]327
// C++ code for Pascal's Triangle
11 C[line, i] = line! / [ [line-i]! * i! ]329
// C++ code for Pascal's Triangle
11 C[line, i] = line! / [ [line-i]! * i! ]331
1 1 1 1 2 1 1 3 3 1 1 4 6 4 193
C[line, i] = line! / [ [line-i]! * i! ]280
C[line, i] = line! / [ [line-i]! * i! ]334
// C++ code for Pascal's Triangle
55C[line, i] = line! / [ [line-i]! * i! ]336
// C++ code for Pascal's Triangle
11C[line, i] = line! / [ [line-i]! * i! ]338
// C++ code for Pascal's Triangle
92
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11____1562_______52
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
using
541 1 1 1 2 1 1 3 3 1 1 4 6 4 11____1562_______56
1 1 1 1 2 1 1 3 3 1 1 4 6 4 193
C[line, i] = line! / [ [line-i]! * i! ]280
C[line, i] = line! / [ [line-i]! * i! ]334_______1560_______55
C[line, i] = line! / [ [line-i]! * i! ]336
// C++ code for Pascal's Triangle
11C[line, i] = line! / [ [line-i]! * i! ]352
C[line, i] = line! / [ [line-i]! * i! ]280
C[line, i] = line! / [ [line-i]! * i! ]334
// C++ code for Pascal's Triangle
55 C[line, i] = line! / [ [line-i]! * i! ]356
// C++ code for Pascal's Triangle
11 C[line, i] = line! / [ [line-i]! * i! ]358
C[line, i] = line! / [ [line-i]! * i! ]151
C[line, i] = line! / [ [line-i]! * i! ]280
C[line, i] = line! / [ [line-i]! * i! ]334
// C++ code for Pascal's Triangle
55 C[line, i] = line! / [ [line-i]! * i! ]356
// C++ code for Pascal's Triangle
11C[line, i] = line! / [ [line-i]! * i! ]365
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
// C++ code for Pascal's Triangle
83 C[line, i] = line! / [ [line-i]! * i! ]280
C[line, i] = line! / [ [line-i]! * i! ]334
// C++ code for Pascal's Triangle
55C[line, i] = line! / [ [line-i]! * i! ]336
// C++ code for Pascal's Triangle
11C[line, i] = line! / [ [line-i]! * i! ]373
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
// C++ code for Pascal's Triangle
0C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
// C++ code for Pascal's Triangle
83 1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 129
1 1 1 1 2 1 1 3 3 1 1 4 6 4 184
C[line, i] = line! / [ [line-i]! * i! ]388
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 184
C[line, i] = line! / [ [line-i]! * i! ]67
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]393
C[line, i] = line! / [ [line-i]! * i! ]394
#include
09
Javascript
#include
10
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]398
namespace
62
namespace
63
C[line, i] = line! / [ [line-i]! * i! ]401
using
81C[line, i] = line! / [ [line-i]! * i! ]07
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 182
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time57
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
namespace
91
C[line, i] = line! / [ [line-i]! * i! ]409
C[line, i] = line! / [ [line-i]! * i! ]1
namespace
03
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
C[line, i] = line! / [ [line-i]! * i! ]413
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]007
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
C[line, i] = line! / [ [line-i]! * i! ]419
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
451 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
6 using
48C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
using
501 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
using
561 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]030
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
namespace
341 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]436
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
48C[line, i] = line! / [ [line-i]! * i! ]441
C[line, i] = line! / [ [line-i]! * i! ]67
// C++ code for Pascal's Triangle
0
// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]446
C[line, i] = line! / [ [line-i]! * i! ]1
#include
99
Đầu ra
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1
Phương pháp này có thể được tối ưu hóa để sử dụng thêm không gian O[n] vì chúng tôi chỉ cần các giá trị từ hàng trước đó. Vì vậy, chúng ta có thể tạo một mảng phụ có kích thước n và ghi đè lên các giá trị. Sau đây là một phương pháp khác chỉ sử dụng không gian thừa O[1]
Phương pháp 3 [ O[n^2] thời gian và O[1] không gian thêm]
Phương pháp này dựa trên phương pháp 1. Chúng ta biết rằng mục thứ i trong một dòng số là Hệ số nhị thức C[dòng, i] và tất cả các dòng bắt đầu bằng giá trị 1. Ý tưởng là tính toán C[dòng, i] bằng cách sử dụng C[dòng, i-1]. Nó có thể được tính trong thời gian O[1] bằng cách sử dụng như sau.
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time
C++
using
00
C[line, i] = line! / [ [line-i]! * i! ]450
C[line, i] = line! / [ [line-i]! * i! ]451
using
03
C[line, i] = line! / [ [line-i]! * i! ]1
using
namespace
C[line, i] = line! / [ [line-i]! * i! ]0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15_______4_______4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
namespace
3
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]466
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]470
C[line, i] = line! / [ [line-i]! * i! ]471
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]476
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]481
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]483
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
C[line, i] = line! / [ [line-i]! * i! ]485
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]487
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]491
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
// C++ code for Pascal's Triangle
0
// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 129
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]01
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
using
811 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]07
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
namespace
5 C[line, i] = line! / [ [line-i]! * i! ]10
// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]511
C
using
90
C[line, i] = line! / [ [line-i]! * i! ]450
C[line, i] = line! / [ [line-i]! * i! ]451
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15_______4_______4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]466
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]470
C[line, i] = line! / [ [line-i]! * i! ]471
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]476
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]57
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]59
C[line, i] = line! / [ [line-i]! * i! ]540
C[line, i] = line! / [ [line-i]! * i! ]481
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]487
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]57
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
C[line, i] = line! / [ [line-i]! * i! ]67
// C++ code for Pascal's Triangle
0
// C++ code for Pascal's Triangle
0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 129
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]01
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
C[line, i] = line! / [ [line-i]! * i! ]4
using
811 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]07
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
namespace
5 C[line, i] = line! / [ [line-i]! * i! ]10
// C++ code for Pascal's Triangle
0
Java
C[line, i] = line! / [ [line-i]! * i! ]565
C[line, i] = line! / [ [line-i]! * i! ]566
namespace
63
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 122
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 123
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 125
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 126
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]573
1 1 1 1 2 1 1 3 3 1 1 4 6 4 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15_______4_______4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 17
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 149
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ]587
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]593
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
C[line, i] = line! / [ [line-i]! * i! ]471
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 162
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 164
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]605
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]481
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]609
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
C[line, i] = line! / [ [line-i]! * i! ]67
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]487
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 173
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 129
1 1 1 1 2 1 1 3 3 1 1 4 6 4 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
namespace
731 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
1 1 1 1 2 1 1 3 3 1 1 4 6 4 139
namespace
771 1 1 1 2 1 1 3 3 1 1 4 6 4 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]07
C[line, i] = line! / [ [line-i]! * i! ]634
// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]264
C[line, i] = line! / [ [line-i]! * i! ]637
Python3
C[line, i] = line! / [ [line-i]! * i! ]638
C[line, i] = line! / [ [line-i]! * i! ]639
C[line, i] = line! / [ [line-i]! * i! ]640
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]642
1 1 1 1 2 1 1 3 3 1 1 4 6 4 157
C[line, i] = line! / [ [line-i]! * i! ]644
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 166
1 1 1 1 2 1 1 3 3 1 1 4 6 4 167
1 1 1 1 2 1 1 3 3 1 1 4 6 4 168
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ]653
1 1 1 1 2 1 1 3 3 1 1 4 6 4 187
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ]656
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]658
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
C[line, i] = line! / [ [line-i]! * i! ]662
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 181
1 1 1 1 2 1 1 3 3 1 1 4 6 4 167
1 1 1 1 2 1 1 3 3 1 1 4 6 4 168
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 186
1 1 1 1 2 1 1 3 3 1 1 4 6 4 187
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 196
C[line, i] = line! / [ [line-i]! * i! ]656
1 1 1 1 2 1 1 3 3 1 1 4 6 4 104
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11____4_______676
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11____4_______678
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
1 1 1 1 2 1 1 3 3 1 1 4 6 4 191
C[line, i] = line! / [ [line-i]! * i! ]681
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
C[line, i] = line! / [ [line-i]! * i! ]684
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
C[line, i] = line! / [ [line-i]! * i! ]658
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]689
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time34
C[line, i] = line! / [ [line-i]! * i! ]691
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time14
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time37
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time42
C[line, i] = line! / [ [line-i]! * i! ]695
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 191
C[line, i] = line! / [ [line-i]! * i! ]698
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]700
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
namespace
77C[line, i] = line! / [ [line-i]! * i! ]485
C[line, i] = line! / [ [line-i]! * i! ]07
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]707
C#
C[line, i] = line! / [ [line-i]! * i! ]708
C[line, i] = line! / [ [line-i]! * i! ]566
C[line, i] = line! / [ [line-i]! * i! ]710
using
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time63
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 125
C[line, i] = line! / [ [line-i]! * i! ]185
C[line, i] = line! / [ [line-i]! * i! ]605
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]717
1 1 1 1 2 1 1 3 3 1 1 4 6 4 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15_______4_______4
C[line, i] = line! / [ [line-i]! * i! ]723
C[line, i] = line! / [ [line-i]! * i! ]605
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]729
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11_______4_______731
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]605
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]470
C[line, i] = line! / [ [line-i]! * i! ]738
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]476
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]605
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]747
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]749
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]751
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
C[line, i] = line! / [ [line-i]! * i! ]684
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]487
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]634
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]759
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
C[line, i] = line! / [ [line-i]! * i! ]761
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]634
C[line, i] = line! / [ [line-i]! * i! ]634
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]766
1 1 1 1 2 1 1 3 3 1 1 4 6 4 131
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 133
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 14
C[line, i] = line! / [ [line-i]! * i! ]770
C[line, i] = line! / [ [line-i]! * i! ]605
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]4
C[line, i] = line! / [ [line-i]! * i! ]774
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]776
C[line, i] = line! / [ [line-i]! * i! ]634
C[line, i] = line! / [ [line-i]! * i! ]634
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]393
C[line, i] = line! / [ [line-i]! * i! ]781
PHP
1 1 1 1 2 1 1 3 3 1 1 4 6 4 177
C[line, i] = line! / [ [line-i]! * i! ]783
C[line, i] = line! / [ [line-i]! * i! ]566
C[line, i] = line! / [ [line-i]! * i! ]710
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]717
1 1 1 1 2 1 1 3 3 1 1 4 6 4 182
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 184
C[line, i] = line! / [ [line-i]! * i! ]131
C[line, i] = line! / [ [line-i]! * i! ]605
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
// C++ code for Pascal's Triangle
55 1 1 1 1 2 1 1 3 3 1 1 4 6 4 191
// C++ code for Pascal's Triangle
55 // C++ code for Pascal's Triangle
771 1 1 1 2 1 1 3 3 1 1 4 6 4 184
1 1 1 1 2 1 1 3 3 1 1 4 6 4 18
// C++ code for Pascal's Triangle
55C[line, i] = line! / [ [line-i]! * i! ]803
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]605
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]807
1 1 1 1 2 1 1 3 3 1 1 4 6 4 191
C[line, i] = line! / [ [line-i]! * i! ]738
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
// C++ code for Pascal's Triangle
11 1 1 1 1 2 1 1 3 3 1 1 4 6 4 191
// C++ code for Pascal's Triangle
11 // C++ code for Pascal's Triangle
77// C++ code for Pascal's Triangle
551 1 1 1 2 1 1 3 3 1 1 4 6 4 18
// C++ code for Pascal's Triangle
11C[line, i] = line! / [ [line-i]! * i! ]803
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]605
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11_______4_______824
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11_______4_______749
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
1 1 1 1 2 1 1 3 3 1 1 4 6 4 191
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
C[line, i] = line! / [ [line-i]! * i! ]807
C[line, i] = line! / [ [line-i]! * i! ]831
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13_______4_______684
1 1 1 1 2 1 1 3 3 1 1 4 6 4 11
C[line, i] = line! / [ [line-i]! * i! ]807
1 1 1 1 2 1 1 3 3 1 1 4 6 4 196
C[line, i] = line! / [ [line-i]! * i! ]807
C[line, i] = line! / [ [line-i]! * i! ]838
// C++ code for Pascal's Triangle
55 C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time14
// C++ code for Pascal's Triangle
11C[line, i] = line! / [ [line-i]! * i! ]842
// C++ code for Pascal's Triangle
11C[line, i] = line! / [ [line-i]! * i! ]485
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]634
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
1 1 1 1 2 1 1 3 3 1 1 4 6 4 191
1 1 1 1 2 1 1 3 3 1 1 4 6 4 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 17
C[line, i] = line! / [ [line-i]! * i! ]684
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]634
C[line, i] = line! / [ [line-i]! * i! ]634
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]766
1 1 1 1 2 1 1 3 3 1 1 4 6 4 184
C[line, i] = line! / [ [line-i]! * i! ]858
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 15
1 1 1 1 2 1 1 3 3 1 1 4 6 4 184
C[line, i] = line! / [ [line-i]! * i! ]67
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]863
#include
09
Javascript
#include
10
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]867
C[line, i] = line! / [ [line-i]! * i! ]566
namespace
63
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]573
1 1 1 1 2 1 1 3 3 1 1 4 6 4 182
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time57
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
C[line, i] = line! / [ [line-i]! * i! ]877
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 18
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 140
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]401
C[line, i] = line! / [ [line-i]! * i! ]883
C[line, i] = line! / [ [line-i]! * i! ]471
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
1 1 1 1 2 1 1 3 3 1 1 4 6 4 14
C[line, i] = line! / [ [line-i]! * i! ]887
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
C[line, i] = line! / [ [line-i]! * i! ]605
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]481
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]893
1 1 1 1 2 1 1 3 3 1 1 4 6 4 13
C[line, i] = line! / [ [line-i]! * i! ]67
C[line, i] = line! / [ [line-i]! * i! ] C[line, i-1] = line! / [ [line - i + 1]! * [i-1]! ] We can derive following expression from above two expressions. C[line, i] = C[line, i-1] * [line - i + 1] / i So C[line, i] can be calculated from C[line, i-1] in O[1] time0
C[line, i] = line! / [ [line-i]! * i! ]487
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
01 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
#include
48C[line, i] = line! / [ [line-i]! * i! ]441
C[line, i] = line! / [ [line-i]! * i! ]67
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 19
// C++ code for Pascal's Triangle
0// C++ code for Pascal's Triangle
0
C[line, i] = line! / [ [line-i]! * i! ]1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 129
C[line, i] = line! / [ [line-i]! * i! ]401
using
81C[line, i] = line! / [ [line-i]! * i! ]07
C[line, i] = line! / [ [line-i]! * i! ]1
C[line, i] = line! / [ [line-i]! * i! ]446
C[line, i] = line! / [ [line-i]! * i! ]1
#include
99
Đầu ra
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1
Thời gian phức tạp. O[n2]
Không gian phụ trợ. Ô[1]
Vì vậy, phương pháp 3 là phương pháp tốt nhất trong số tất cả, nhưng nó có thể gây tràn số nguyên cho các giá trị lớn của n vì nó nhân hai số nguyên để thu được các giá trị.
Các biến thể của vấn đề có thể được hỏi trong các cuộc phỏng vấn
i] Tìm toàn bộ tam giác pascal như hình trên
ii] Tìm chỉ một phần tử của tam giác pascal đã cho số hàng và số cột trong thời gian O[n]