Double factorial

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: $n!!={\begin{cases}1&{\text{if }}n=0{\text{ or }}n=1\\n\times (n-2)!!&{\text{if }}n\geq 2\qquad \qquad \end{cases}}$ The double factorial is not defined when n is a negative even integer.

Do not confuse the double factorial for a factorial computed twice.

$a!!\neq (a!)!$

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.

$a!=a(a-1)(a-2)\cdots 3\cdot 2\cdot 1$

$a!!=a(a-2)(a-4)(a-6)\cdots$

The double factorial terminates with the sequence of evens, for example: $4\cdot 2\cdot 0!!$ or the sequence of odds: eg $5\cdot 3\cdot 1!!$

where $1!!=0!!=1$

The following properties hold:

$a!!=a\cdot (a-2)!!$

$a!=a!!\cdot (a-1)!!$

$(2a)!!=a!\cdot 2^{a}$ 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