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 $$\mathit{sf}(n)$$ may be defined as: $$\begin{align} \mathit{sf}(n) &= 1!\cdot 2!\cdot \cdots n! = \prod_{i=1}^{n} i! = n!\cdot\mathit{sf}(n-1)\\ &= 1^n \cdot 2^{n-1} \cdot \cdots n = \prod_{i=1}^{n} i^{n+1-i}.\\ \end{align}$$ Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence of superfactorials, beginning with $$\mathit{sf}(0)=1$$, 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 $$\mathit{sf}(p-1)\equiv(p-1)!!\pmod{p},$$ where $$!!$$ is the notation for the double factorial.

For every integer $$k$$, the number $$\mathit{sf}(4k)/(2k)!$$ is a square number. This may be expressed as stating that, in the formula for $$\mathit{sf}(4k)$$ 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 $$\mathit{sf}(n)$$, and equals the superfactorial when the given numbers are consecutive.