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));
}
}
|
PHP
<?php
function doublefactorial($n)
{
if ($n == 0 || $n==1)
return 1;
return $n * doublefactorial($n - 2);
}
echo "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>
|
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));
}
}
|
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);
?>
|
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>
|
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!! = 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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