Superfactorial

In mathematics, and more specifically number theory, the superfactorial of a positive integer $n$ is the product of the first $n$ factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.

Definition

The $n$th superfactorial $\mathit(n)$ may be defined as:
$\begin \mathit(n) &= 1!\cdot 2!\cdot \cdots n! = \prod_^ i! = n!\cdot\mathit(n-1)\\ &= 1^n \cdot 2^ \cdot \cdots n = \prod_^ i^.\\ \end$
Following the usual convention for the empty product, the superfactorial of 0 is 1. The integer sequence of superfactorials, beginning with $\mathit(0)=1$, is:

Properties

Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function. According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when $p$ is an odd prime number
$\mathit(p-1)\equiv(p-1)!!\pmod,$
where the $!!$ is the notation for the double factorial.

For every integer $k$, the number $\mathit(4k)/(2k)!$ is a square number. This may be expressed as stating that, in the formula for $\mathit(4k)$ as a product of factorials, omitting one of the factorials (the middle one, $(2k)!$) results in a square product. Additionally, if any $n+1$ integers are given, the product of their pairwise differences is always a multiple of $\mathit(n)$, and equals the superfactorial when the given numbers are consecutive.

References

External links

*{{MathWorld, id=Superfactorial, title=Superfactorial, mode=cs2

Integer sequences
Factorial and binomial topics