Table of integer values

 

DOUBLE FACTORIALS

n!! denotes the double factorial of n and is defined recursively by



  n!!=

  \begin{cases}

    1,&\text{if }n=0\text{ or }n=1;

   \\

    n\times(n-2)!! &\text{if }n\ge2.\qquad\qquad

  \end{cases}

For example, 8!! 945. The sequence of double factorials for n = 0, 1, 2, ... starts as

1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, ...

The above definition can be rewritten to define double factorials of negative odd numbers:

 (n-2)!!=\frac{n!!}{n}.

The sequence of double factorials for n = −1, −3, −5, −7, ... starts as

1, -1, \frac{1}{3}, -\frac{1}{15}, \dots

while the double factorial of negative even integers is undefined.

Some identities involving double factorials are:



  n!!=

  \begin{cases}

    2^{n/2} \cdot \frac{n}{2}!& \text{if }n\mbox{ is even,}\qquad\qquad \\

    \dbinom{\frac{n}{2} }{ \frac{n-1}{2}} \cdot 2^{(n-1)/2} \cdot \frac{n-1}{2}! & \text{if }n\text{ is odd}

  \end{cases}

n!=n!!(n-1)!! \,

(2n)!!=2^nn! \,

(2n+1)!!={(2n+1)!\over(2n)!!}={(2n+1)!\over2^nn!}

(2n-1)!!={(2n-1)!\over(2n-2)!!}={(2n)!\over2^nn!}

\Gamma\left(n+{1\over2}\right)=\sqrt{\pi}\,\,{(2n-1)!!\over2^n}

\Gamma\left({n\over2}+1\right)=\sqrt{\pi}\,\,{n!!\over2^{(n+1)/2}}

where Γ is the Gamma function. The last equation above can be used to define the double factorial as a function of any complex number n ≠ 0, just as the Gamma function generalizes the factorial function. One should be careful not to interpret n!! as the factorial of n!, which would be written (n!)! and is a much larger number (for n > 2).

 

MULTIFACTORIALS

A common related notation is to use multiple exclamation points to denote a multifactorial, the product of integers in steps of two (n!!), three (n!!!), or more. The double factorial is the most commonly used variant, but one can similarly define the triple factorial (n!!!) and so on. In general, the kth factorial, denoted by n!(k), is defined recursively as



  n!^{(k)}=

  \left\{

   \begin{matrix}

    1,\qquad\qquad\ &&\mbox{if }0\le n<k;

   \\

    n(n-k)!^{(k)},&&\mbox{if }n\ge k.\quad\ \ \,

   \end{matrix}

  \right.

Some mathematicians have suggested an alternative notation of n!2 for the double factorial and similarly n!k for other multifactorials, but this has not come into general use.

Note that ( − n)! is not defined for integers, and ( − n)!! is not defined for even integers, ( − n)!(k) is not defined for

 n\equiv 0(\mbox{mod k}).

Also, (kn)!(k) = knn!

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