Dado un entero positivo n . La tarea es encontrar la suma del coeficiente binomial incluso indexado. Es decir,
n C 0 + n C 2 + n C 4 + n C 6 + n C 8 + ………..
Ejemplos:
Input : n = 4 Output : 8 4C0 + 4C2 + 4C4 = 1 + 6 + 1 = 8 Input : n = 6 Output : 32
Método 1: (Fuerza bruta)
La idea es encontrar todos los coeficientes binomiales y encontrar solo la suma de los valores indexados pares.
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.
?>
Javascript
<script>
// Javascript Program to find sum
// of even index term
// Return the sum of
// even index term
function evenSum(n)
{
var C = Array.from(Array(n+1),
()=> Array(n+1).fill(0));
var 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;
// 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.
var sum = 0;
for (var i = 0; i <= n; i += 2)
sum += C[n][i];
return sum;
}
// Driver Program
var n = 4;
document.write( evenSum(n) );
</script>
Producción :
8
Complejidad de tiempo: O(n 2 )
Método 2: (usando la fórmula)
Suma del coeficiente binomial incluso indexado: 
Prueba:
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 + .......
A continuación se muestra la implementación de este enfoque:
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));
}
/* Driver 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.
?>
Javascript
<script>
// JavaScript 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
let n = 4;
document.write(evenbinomialCoeffSum(n));
// This code is contributed by code_hunt.
</script>
Producción:
8
Complejidad temporal: O(1)
Suma del coeficiente binomial de índice impar
Usando el resultado anterior, podemos demostrar fácilmente que la suma del coeficiente binomial de índice impar también es 2 n-1 .