Double factorial - GeeksforGeeks

29 Apr, 2021
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++
Java
Python3
C#
PHP
Javascript
#include<stdio.h>  // function to find double factorial of given numberunsigned 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 numberdef doublefactorial(n):     if (n == 0 or n == 1):        return 1;    return n * doublefactorial(n - 2); # Driver Codeprint("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.
<?php// PHP code for// Double factorial // function return// double factorialfunction 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.?>
<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>
Output:  
Double factorial is 15
Iterative Solution: Double factorial can also be calculated iteratively as recursion can be costly for large numbers.  
C++
Java
Python3
C#
PHP
Javascript
#include<stdio.h>  // function to find double factorial of given numberunsigned 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 numberimport 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 numberdef doublefactorial(n):    res = 1;    for i in range(n, -1, -2):        if(i == 0 or i == 1):            return res;        else:            res *= i;     # Driver Codeprint("Double factorial is",        doublefactorial(5)); # This code is contributed by mits
// C# Program to find double factorial// of given numberusing 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 // function to find double// factorial of given numberfunction 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.?>
<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>
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.  
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_factorialThis article is contributed by Rahul Agrawal. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. 
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