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. One can define the k-th factorial, denoted by n!(k), recursively for non-negative integers as:

$n\equiv 0(\mbox{mod k})$

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. With the above definition, $n\equiv 0(\mbox{mod k})$

In the same way that n! is not defined for negative integers, and n!! is not defined for negative even integers, n!k is not defined for negative integers evenly divisible by k.

ome 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. With the above definition.

ome 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. With the above definition.