中央二项式系数

给定整数N ，任务是找到$N^{th}$中央二项式系数。
N = 0、1、2、3 ..的前几个中央二项式系数为

1, 2, 6, 20, 70, 252, 924, 3432…..

例子：

Input: N = 3
Output: 20
Explanation:
$N^{th}$ Central Binomial Cofficient = $\binom{2N}{N}$ = $\binom{2*3}{3}$ = $\frac{6*5*4}{3*2*1}$ = 20

Input: N = 2
Output: 6

方法：中心二项式系数是以下形式的二项式系数： $\binom{2N}{N}$ 。二项式系数$\binom{2N}{N}$可以使用这种方法通过动态编程针对给定值N计算该值。

例如：

Central binomial coefficient of N = 3 is given by:
$\binom{2N}{N}$ = $\binom{2*3}{3}$ = $\frac{6*5*4}{3*2*1}$ = 20

下面是上述方法的实现：

C++

// C++ implementation to find the 
// Nth Central Binomial Coefficient
  
#include 
using namespace std; 
  
// Function to find the value of 
// Nth Central Binomial Coefficient
int binomialCoeff(int n, int k) 
{ 
    int C[n + 1][k + 1]; 
    int i, j; 
  
    // Calculate value of Binomial
    // Coefficient in bottom up manner 
    for (i = 0; i <= n; i++) 
    { 
        for (j = 0; j <= min(i, k); j++) 
        { 
            // Base Cases 
            if (j == 0 || j == i) 
                C[i][j] = 1; 
  
            // Calculate value 
            // using previously 
            // stored values 
            else
                C[i][j] = C[i - 1][j - 1] + 
                        C[i - 1][j]; 
        } 
    } 
  
    return C[n][k]; 
} 
  
// Driver Code 
int main() 
{ 
    int n = 3;
    int k = n;
    n = 2*n;
    cout << binomialCoeff(n, k); 
}

Java

// Java implementation to find the 
// Nth Central Binomial Coefficient
class GFG{
      
// Function to find the value of 
// Nth Central Binomial Coefficient
static int binomialCoeff(int n, int k) 
{ 
    int[][] C = new int[n + 1][k + 1]; 
    int i, j; 
  
    // Calculate value of Binomial
    // Coefficient in bottom up manner 
    for(i = 0; i <= n; i++) 
    { 
       for(j = 0; j <= Math.min(i, k); j++) 
       { 
             
          // Base Cases 
          if (j == 0 || j == i) 
              C[i][j] = 1; 
            
          // Calculate value 
          // using previously 
          // stored values 
          else
              C[i][j] = C[i - 1][j - 1] + 
                        C[i - 1][j]; 
       } 
    } 
    return C[n][k]; 
} 
  
// Driver Code 
public static void main(String[] args)
{ 
    int n = 3;
    int k = n;
    n = 2 * n;
      
    System.out.println(binomialCoeff(n, k)); 
}
}
  
// This code is contributed by Ritik Bansal

Python3

# C# implementation to find the 
# Nth Central Binomial Coefficient
  
# Function to find the value of 
# Nth Central Binomial Coefficient
def binomialCoeff(n, k):
      
    C = [[0 for j in range(k + 1)] 
            for i in range(n + 1)]
      
    i = 0
    j = 0
      
    # Calculate value of Binomial
    # Coefficient in bottom up manner
    for i in range(n + 1):
        for j in range(min(i, k) + 1):
              
            # Base Cases
            if j == 0 or j == i:
                C[i][j] = 1
                  
            # Calculate value 
            # using previously 
            # stored values 
            else:
                C[i][j] = (C[i - 1][j - 1] +
                           C[i - 1][j])
      
    return C[n][k]
      
# Driver code
if __name__=='__main__':
      
    n = 3
    k = n
    n = 2 * n
      
    print(binomialCoeff(n, k))
          
# This code is contributed by rutvik_56

C#

// C# implementation to find the 
// Nth Central Binomial Coefficient
using System;
class GFG{
      
// Function to find the value of 
// Nth Central Binomial Coefficient
static int binomialCoeff(int n, int k) 
{ 
    int [,]C = new int[n + 1, k + 1]; 
    int i, j; 
  
    // Calculate value of Binomial
    // Coefficient in bottom up manner 
    for(i = 0; i <= n; i++) 
    { 
       for(j = 0; j <= Math.Min(i, k); j++)
       {
             
          // Base Cases 
          if (j == 0 || j == i) 
              C[i, j] = 1; 
                
          // Calculate value 
          // using previously 
          // stored values 
          else
              C[i, j] = C[i - 1, j - 1] + 
                        C[i - 1, j]; 
       } 
    } 
    return C[n, k]; 
} 
  
// Driver Code 
public static void Main()
{ 
    int n = 3;
    int k = n;
    n = 2 * n;
      
    Console.Write(binomialCoeff(n, k)); 
}
}
  
// This code is contributed by Code_Mech

输出：

20