Double Factorial

The double factorial (also called semifactorial) of a number $ n $, often denoted $ n!! $, is a mathematical operation applied to a positive integer $ n $ which consists of the multiplication (the product) of all non-zero integers less than or equal to $ n $ which have the same parity as $ n $.

The formula for the double factorial is: $$ n!! = \prod_{k=0}^{\left\lceil\frac{n}{2}\right\rceil - 1} (n-2k) = n (n-2) (n-4) \cdots $$

If $ n $ is an even number (multiple of 2) then $ n!! $ is the multiplication of all multiples of $ 2 $ less or equal than $ n $ (and greater than $ 0 $)

If $ n $ is an odd number (not a multiple of 2) then $ n!! $ is the multiplication of all non-multiple numbers of $ 2 $ less or equal than $ n $ (and greater than $ 0 $).

By convention, the double factorial of zero is equal to 1: $ 0!! = 1 $

The values of the first double factorials: $$ 0!! = 1 \\ 1!! = 1 \\ 2!! = 2 \\ 3!! = 3 \\ 4!! = 8 \\ 5!! = 15 \\ 6!! = 48 \\ 7!! = 105 \\ 8!! = 384 \\ 9!! = 945 \\ 10!! = 3840 $$

The remarkable relations of the double factorial with the factorial are:

$$ n! = n!! (n-1)!! $$

$$ n!! = \frac{n!}{(n-1)!!} = \frac{(n+1)!}{(n+1)!!} $$

When the two exclamation points are to the left of the number, it may be the subfactorial or the double subfactorial.

$$ !n = n!\sum_{k=0}^n \frac{(-1)^k}{k!} $$

$$ !!n= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor }\,n!! \sum_{i=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \frac{(-1)^i}{(n-2 i)!!} $$

A non-recursive function to calculate the double factorial of a number N is: // Pseudo-code
function doubleFactorial(n) {
if (n == 0 OR n == 1) return 1
result = 1
for i from n down to 2 by 2 {
result = result * i
}
return result
}