The double factorial is an extension onto the normal factorial function. It is denoted with two exclamation points: $ a!! $ .
|Definition: Double factorial|
| The double factorial of an integer $ n $ is defined recursively as:
Do not confuse the double factorial for a factorial computed twice.
- $ a!!\ne(a!)! $
The double in double factorial represents the increment between the values of the terms when the factorial is expanded into a product. In the case of a regular factorial, each factor is decremented by one, from the number 'a' to 1. In the case of a double factorial, each factor is decremented by two.
- $ a!=a(a-1)(a-2)\cdots3\cdot2\cdot1 $
- $ a!!=a(a-2)(a-4)(a-6)\cdots $
The double factorial terminates with the sequence of evens, for example: $ 4\cdot2\cdot0!! $ or the sequence of odds: eg $ 5\cdot3\cdot1!! $
where $ 1!!=0!!=1 $
The following properties hold:
- $ a!!=a\cdot(a-2)!! $
- $ a!=a!!\cdot(a-1)!! $
- $ (2a)!!=a!\cdot2^a $ for any integer $ a $
There also exists the triple factorial, which is not as commonly known or used as the double, and with it a set of of analogous properties.
- A006882 - Double factorials $ n!!:a(n)=n\cdot a(n-2) $ in OEIS