# 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:

For odd n, the double factorial is:

## 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 C implementation of double factorial.

```#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;
}
```

Output:

```Double factorial is 15
```

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

```#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;
}

```

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.

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.

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