Double factorial of a non-negative integer n, is the product of all the integers from 1 to n that have the same parity (odd or even) as n. It is also called as semifactorial of a number and is denoted by !!. For example, double factorial of 9 is 9*7*5*3*1 which is 945. Note that, a consequence of this definition is 0!! = 1.
Examples:
Input: 6 Output: 48 Note that 6*4*2 = 48 Input: 7 Output: 105 Note that 7*5*3 = 105
For even n, the double factorial is:
For odd n, the double factorial is:
Recursive Solution:
Double factorial can be calculated using following recursive formula.
n!! = n * (n-2)!! n!! = 1 if n = 0 or n = 1
Following is the implementation of double factorial.
C++
#include<stdio.h> // function to find double factorial of given number unsigned int doublefactorial(unsigned int n) { if (n == 0 || n==1) return 1; return n*doublefactorial(n-2); } int main() { printf ( "Double factorial is %d" , doublefactorial(5)); return 0; } |
Java
import java.io.*; class GFG { // function to find double factorial // of given number static long doublefactorial( long n) { if (n == 0 || n== 1 ) return 1 ; return n * doublefactorial(n - 2 ); } // Driver code static public void main (String[] args) { System.out.println( "Double factorial" + " is " + doublefactorial( 5 )); } } // This code is contributed by anuj_67. |
Python3
# function to find double # factorial of given number def doublefactorial(n): if (n = = 0 or n = = 1 ): return 1 ; return n * doublefactorial(n - 2 ); # Driver Code print ( "Double factorial is" , doublefactorial( 5 )); # This code is contributed # by Smitha |
C#
using System; class GFG { // function to find double factorial // of given number static uint doublefactorial( uint n) { if (n == 0 || n==1) return 1; return n * doublefactorial(n - 2); } // Driver code static public void Main () { Console.WriteLine( "Double factorial" + " is " + doublefactorial(5)); } } // This code is contributed by anuj_67. |
PHP
<?php // PHP code for // Double factorial // function return // double factorial function doublefactorial( $n ) { if ( $n == 0 || $n ==1) return 1; return $n * doublefactorial( $n - 2); } // Driver Code echo "Double factorial is " , doublefactorial(5); // This code is contributed by Ajit. ?> |
Output:
Double factorial is 15
Iterative Solution:
Double factorial can also be calculated iteratively as recursion can be costly for large numbers.
C++
#include<stdio.h> // function to find double factorial of given number unsigned int doublefactorial(unsigned int n) { int res = 1; for ( int i=n; i>=0; i=i-2) { if (i==0 || i==1) return res; else res *= i; } } int main() { printf ( "Double factorial is %d" , doublefactorial(5)); return 0; } |
Java
// Java Program to find double factorial // of given number import java .io.*; class GFG { // function to find double factorial // of given number static int doublefactorial( int n) { int res = 1 ; for ( int i = n; i >= 0 ; i = i- 2 ) { if (i == 0 || i == 1 ) return res; else res *= i; } return res; } // Driver code public static void main(String[] args) { System.out.println( "Double factorial" + " is " + doublefactorial( 5 )); } } // This code is Contributed by Anuj_67 |
Python3
# Python3 Program to find double # factorial of given number # Function to find double # factorial of given number def doublefactorial(n): res = 1 ; for i in range (n, - 1 , - 2 ): if (i = = 0 or i = = 1 ): return res; else : res * = i; # Driver Code print ( "Double factorial is" , doublefactorial( 5 )); # This code is contributed by mits |
C#
// C# Program to find double factorial // of given number using System; class GFG { // function to find double factorial // of given number static int doublefactorial( int n) { int res = 1; for ( int i = n; i >= 0; i = i-2) { if (i == 0 || i == 1) return res; else res *= i; } return res; } // Driver code static void Main() { Console.Write( "Double factorial" + " is " + doublefactorial(5)); } } // This code is Contributed by Anuj_67 |
PHP
<?php // function to find double // factorial of given number function doublefactorial( $n ) { $res = 1; for ( $i = $n ; $i >= 0; $i = $i - 2) { if ( $i == 0 or $i == 1) return $res ; else $res *= $i ; } } // Driver Code echo "Double factorial is " , doublefactorial(5); // This code is contributed by anuj_67. ?> |
Output:
Double factorial is 15
Time complexity of the above solutions is O(n).
Important Points :
- Double factorial and factorial are related using below formula.
Note : n!! means double factorial. If n is even, i.e., n = 2k n!! = 2kk! Else (n = 2k + 1) n!! = (2k)! / 2kk!
- Double factorial is frequently used in combinatorics. Refer wiki for list of applications. An example application is count of perfect matchings of a complete graph Kn+1 for odd n.
References:
https://en.wikipedia.org/wiki/Double_factorial
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