Double factorial
Last Updated :
07 Jan, 2024
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>
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 {
static long doublefactorial( long n)
{
if (n == 0 || n== 1 )
return 1 ;
return n * doublefactorial(n - 2 );
}
static public void main (String[] args)
{
System.out.println( "Double factorial"
+ " is " + doublefactorial( 5 ));
}
}
|
Python3
def doublefactorial(n):
if (n = = 0 or n = = 1 ):
return 1 ;
return n * doublefactorial(n - 2 );
print ( "Double factorial is" ,
doublefactorial( 5 ));
|
C#
using System;
class GFG {
static uint doublefactorial( uint n)
{
if (n == 0 || n==1)
return 1;
return n * doublefactorial(n - 2);
}
static public void Main ()
{
Console.WriteLine( "Double factorial"
+ " is " + doublefactorial(5));
}
}
|
Javascript
<script>
function doublefactorial(n)
{
if (n == 0 || n==1)
return 1;
return n * doublefactorial(n - 2);
}
document.write( "Double factorial"
+ " is " + doublefactorial(5));
</script>
|
PHP
<?php
function doublefactorial( $n )
{
if ( $n == 0 || $n ==1)
return 1;
return $n * doublefactorial( $n - 2);
}
echo "Double factorial is " ,
doublefactorial(5);
?>
|
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>
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
import java .io.*;
class GFG {
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;
}
public static void main(String[] args)
{
System.out.println( "Double factorial"
+ " is " + doublefactorial( 5 ));
}
}
|
Python3
def doublefactorial(n):
res = 1 ;
for i in range (n, - 1 , - 2 ):
if (i = = 0 or i = = 1 ):
return res;
else :
res * = i;
print ( "Double factorial is" ,
doublefactorial( 5 ));
|
C#
using System;
class GFG {
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;
}
static void Main()
{
Console.Write( "Double factorial"
+ " is " + doublefactorial(5));
}
}
|
Javascript
<script>
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
<?php
function doublefactorial( $n )
{
$res = 1;
for ( $i = $n ; $i >= 0; $i = $i - 2)
{
if ( $i == 0 or $i == 1)
return $res ;
else
$res *= $i ;
}
}
echo "Double factorial is " , doublefactorial(5);
?>
|
Output:
Double factorial is 15
Time complexity: O(n).
Auxiliary Space: O(1) since using constant space
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!
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
Like Article
Suggest improvement
Share your thoughts in the comments
Please Login to comment...