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:
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; } |
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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. |
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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 |
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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. |
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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. ?> |
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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; } |
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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 |
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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 |
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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 |
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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. ?> |
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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!
- 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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