Double Factorial

The double factorial (also called semifactorial) of a number $ n $, denoted by $ n!! $ (with 2 exclamation marks), is the multiplication (the product) of all non-zero integers less than or equal to $ n $ that have the same parity.

$$ n!! = \prod_{k=0}^{\left\lceil\frac{n}{2}\right\rceil - 1} (n-2k) = n (n-2) (n-4) \cdots $$

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 $).

Example: $$ 8!! = 2 \times 4 \times 6 \times 8 = 384 $$

Example: $$ 5!! = 1 \times 3 \times 5 = 15 $$

Be careful not to confuse the double factorial $ n !! $ with the factorial of factorial $ (n!)! $

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:

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

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