Double factorial - GeeksforGeeks

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:
$n!!=\prod_{k=1}^{{n+1}/2}(2k-1)=n(n-2)(n-4).....3*1$

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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!! = 2^kk!
Else (n = 2k + 1)
   n!! = (2k)! / 2^kk!

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

This article is contributed by Rahul Agrawal. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.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.

My Personal Notes