The coefficients that appear in the binomial expansion are called binomial coefficients.

Binomial coefficients can be recursively defined as follows:

C(n, 0) = C(n, n) = 1 for all n > 0;

C(n, k) = C(n -1, k -1) + C(n – 1, k) for all 0 < k < n.

Arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascal's triangle – Wikipedia .

Algorithm:

algorithm Binomial(n, k)
// Computes C(n, k) using dynamic programming
// Input: Integers n ≥ k ≥ 0
// Output: The value of C(n, k)
for i ← 0 to n do
for j ← 0 to min(i, k) do
if j = 0 or j = i then
A[i, j] ← 1
else A[i, j] ← A[i − 1, j − 1] + A[i − 1, j]
return A[n, k]

Implementation:

public class pascal {
public static void main(String [] args)
{
int [][] pascal = new int[8][];
for (int i=0; i<pascal.length; i++)
{
pascal[i] = new int[i+1];
pascal[i][0] = 1;
pascal[i][i] = 1;
for (int j=1; j<i; j++)
{
pascal[i][j] = pascal[i-1][j]+pascal[i-1][j-1];
}
}
for (int i=0; i<pascal.length; i++)
{
for (int j=0; j<pascal[i].length; j++)
{
System.out.print(" "+pascal[i][j]);
}
System.out.println("");
}
}
}

> java pascal

1

1 1

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

1 7 21 35 35 21 7 1

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