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Double factorial

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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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