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.

C++

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#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;
}

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Java

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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.

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Python3

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# 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

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C#

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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.

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PHP

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<?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.
?>

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Output:

Double factorial is 15

Iterative Solution:
Double factorial can also be calculated iteratively as recursion can be costly for large numbers.

C++

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#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;
}

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Java

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// 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 

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Python3

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# 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

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C#

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// 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 

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PHP

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<?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.
?>

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Output:

Double factorial is 15

Time complexity of the above solutions is O(n).

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 Kn+1 for odd n.

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

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