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Find sum of even index binomial coefficients

Given a positive integer n. The task is to find the sum of even indexed binomial coefficient. That is,
nC0 + nC2 + nC4 + nC6 + nC8 + ………..

Examples :

Input : n = 4
Output : 8
4C0 + 4C2 + 4C4
= 1 + 6 + 1
= 8

Input : n = 6
Output : 32



Method 1: (Brute Force)
The idea is to find all the binomial coefficient and find only the sum of even indexed value.

CPP

// CPP Program to find sum 
// of even index term
#include <bits/stdc++.h>
using namespace std;
  
// Return the sum of 
// even index term
int evenSum(int n)
{
    int C[n + 1][n + 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, n); 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];
        }
    }    
  
    // finding sum of even index term.
    int sum = 0;
    for (int i = 0; i <= n; i += 2)
        sum += C[n][i];
  
    return sum;
}
  
// Driver Program
int main()
{
    int n = 4;
    cout << evenSum(n) << endl;
    return 0;
}

Java

// Java Program to find sum 
// of even index term
import java.io.*;
import java.math.*;
  
class GFG {
      
    // Return the sum of 
    // even index term
    static int evenSum(int n)
    {
        int C[][] = new int [n + 1][n + 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, n); j++)
            {
                // Base Cases
                if (j == 0 || j == i)
                    C[i][j] = 1;
       
                // else Calculate value using 
                // previously stored values
                else
                    C[i][j] = C[i - 1][j - 1
                                + C[i - 1][j];
            }
        }    
       
        // finding sum of even index term.
        int sum = 0;
        for (i = 0; i <= n; i += 2)
            sum += C[n][i];
       
        return sum;
    }
       
    // Driver Program
    public static void main(String args[])
    {
        int n = 4;
        System.out.println(evenSum(n));
    }
}
  
/*This code is contributed by Nikita Tiwari.*/

Python

# Python Program to find sum of even index term
import math 
  
# Return the sum of even index term
def evenSum(n) :
    # Creates a list containing n+1 lists,
    # each of n+1 items, all set to 0
    C = [[0 for x in range(n + 1)] for y in range(n + 1)] 
  
    # Calculate value of Binomial Coefficient
    # in bottom up manner
    for i in range(0, n + 1):
        for j in range(0, min(i, n + 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]
          
    # Finding sum of even index term
    sum = 0;
    for i in range(0, n + 1):
        if n % 2 == 0:
            sum = sum + C[n][i]
              
    return sum
      
# Driver method
n = 4
print evenSum(n)
  
  
# This code is contributed by 'Gitanjali'.

C#

// C# Program to find sum 
// of even index term
using System;
  
class GFG {
      
    // Return the sum of 
    // even index term
    static int evenSum(int n)
    {
        int [,]C = new int [n + 1,n + 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, n); j++)
            {
                // Base Cases
                if (j == 0 || j == i)
                    C[i,j] = 1;
      
                // else Calculate value using 
                // previously stored values
                else
                    C[i,j] = C[i - 1,j - 1] 
                            + C[i - 1,j];
            }
        
      
        // finding sum of even index term.
        int sum = 0;
        for (i = 0; i <= n; i += 2)
            sum += C[n,i];
      
        return sum;
    }
      
    // Driver Program
    public static void Main()
    {
        int n = 4;
        Console.WriteLine(evenSum(n));
    }
}
  
/*This code is contributed by vt_m.*/

PHP

<?php
// PHP Program to find sum 
// of even index term
  
// Return the sum of 
// even index term
function evenSum($n)
{
    $C = array(array());
    $i; $j;
  
    // Calculate value of Binomial 
    // Coefficient in bottom up manner
    for ($i = 0; $i <= $n; $i++) 
    {
        for ($j = 0; $j <= min($i, $n); $j++)
        {
            // Base Cases
            if ($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];
        }
    
  
    // finding sum of even index term.
    $sum = 0;
    for ( $i = 0; $i <= $n; $i += 2)
        $sum += $C[$n][$i];
  
    return $sum;
}
  
// Driver Code
$n = 4;
echo evenSum($n) ;
  
// This code is contributed by anuj_67.
?>


Output :

8

Time Complexity : O(n2)

Method 2: (Using Formula)
Sum of even indexed binomial coeffient :
^nC_0 + ^nC_2 + ^nC_4 + ^nC_6 + .... = 2^{n-1}

Proof :

We know,
(1 + x)n = nC0 + nC1 x + nC2 x2 + ..... + nCn xn

Now put x = -x, we get
(1 - x)n = nC0 - nC1 x + nC2 x2 + ..... + (-1)n nCn xn

Now, adding both the above equation, we get,
(1 + x)n + (1 - x)n = 2 * [nC0 + nC2 x2 + nC4 x4 + .......]

Put x = 1
(1 + 1)n + (1 - 1)n = 2 * [nC0 + nC2 + nC4 + .......]
2n/2 = nC0 + nC2 + nC4 + .......
2n-1 = nC0 + nC2 + nC4 + .......

Below is the implementation of this approach :

C++

// CPP Program to find sum even indexed Binomial
// Coefficient.
#include <bits/stdc++.h>
using namespace std;
  
// Returns value of even indexed Binomial Coefficient
// Sum which is 2 raised to power n-1.
int evenbinomialCoeffSum(int n)
{
    return (1 << (n - 1));
}
  
/* Drier program to test above function*/
int main()
{
    int n = 4;
    printf("%d", evenbinomialCoeffSum(n));
    return 0;
}

Java

// Java Program to find sum even indexed 
// Binomial Coefficient.
import java.io.*;
  
class GFG {
// Returns value of even indexed Binomial Coefficient
// Sum which is 2 raised to power n-1.
static int evenbinomialCoeffSum(int n)
{
    return (1 << (n - 1));
}
  
// Driver Code
public static void main(String[] args)
{
int n = 4;
    System.out.println(evenbinomialCoeffSum(n));
}
    }
  
// This code is contributed by 'Gitanjali'.

Python

# Python program to find sum even indexed 
# Binomial Coefficient
import math 
  
# Returns value of even indexed Binomial Coefficient
# Sum which is 2 raised to power n-1.
def evenbinomialCoeffSum( n):
  
    return (1 << (n - 1))
  
# Driver method
if __name__ == '__main__':
    n = 4
    print evenbinomialCoeffSum(n)
  
# This code is contributed by 'Gitanjali'.

C#

// C# Program to find sum even indexed 
// Binomial Coefficient.
using System;
  
class GFG 
{
    // Returns value of even indexed 
    // Binomial Coefficient Sum which 
    // is 2 raised to power n-1.
    static int evenbinomialCoeffSum(int n)
    {
        return (1 << (n - 1));
    }
      
    // Driver Code
    public static void Main()
    {
        int n = 4;
        Console.WriteLine(evenbinomialCoeffSum(n));
    }
}
  
// This code is contributed by 'Vt_m'.

PHP

<?php
// PHP Program to find sum 
// even indexed Binomial
// Coefficient.
  
// Returns value of even indexed
// Binomial Coefficient Sum which 
// is 2 raised to power n-1.
function evenbinomialCoeffSum( $n)
{
    return (1 << ($n - 1));
}
  
    // Driver Code
    $n = 4;
    echo evenbinomialCoeffSum($n);
  
// This code is contributed by anuj_67.
?>


Output :

8

Time Complexity : O(1)

Sum of odd index binomial coefficient
Using the above result we can easily prove that the sum of odd index binomial coefficient is also 2n-1.



This article is attributed to GeeksforGeeks.org

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