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 nth 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}\\ &=\frac{(n!)^{n+1}}{\prod_{i=1}^{n} i^{i}} = \frac{(n!)^{n+1}}{H(n)} \end{align}where H is the hyperfactorial.