Double factorial

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:
n!!=\prod_{k=1}^{n/2}(2k)=n(n-2)(n-4).....4*2      
For odd n, the double factorial is:
n!!=\prod_{k=1}^{{n+1}/2}(2k-1)=n(n-2)(n-4).....3*1      
 

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. 
 

#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;
}

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.

# 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

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.

<script>
// Javascript program to find double factorial of given number

    // function to find double factorial
    // of given number
    function doublefactorial(n)
    {
        if (n == 0 || n==1)
            return 1;    
        return n * doublefactorial(n - 2);
    }
     
// Driver code    

    document.write("Double factorial"
            + " is " + doublefactorial(5));
          
// This code is contributed by susmitakundugoaldanga.          
</script>

<?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. 
 

#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 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 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# 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 

<script>

    // Javascript Program to find
    // double factorial of given number
    
    // function to find double factorial
    // of given number
    function doublefactorial(n)
    {
        let res = 1;
        for (let i = n; i >= 0; i = i-2)
        {
            if (i == 0 || i == 1)
                return res;
            else
                res *= i;
        }
         
        return res;
    }
    
    document.write("Double factorial" + " is " 
    + doublefactorial(5));
    
</script>

<?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: O(n).

Auxiliary Space: O(1) since using constant space

Important Points : 
 

1. 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! 

2.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 K_n+1 for odd n. 
 

References: 
https://en.wikipedia.org/wiki/Double_factorial