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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:
$ n!!=\begin{cases}1&\text{if }n=0\text{ or }n=1 \\ n\times(n-2)!!&\text{if }n\ge2\qquad\qquad\end{cases} $

The double factorial is not defined when n is a negative even integer.

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.

See also

  • A006882 - Double factorials $ n!!:a(n)=n\cdot a(n-2) $ in OEIS
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