# Superfactorial

The factorial of a positive integer n is the product of the numbers from 1 up to and including n: n! = 1 × 2 × 3 × … × n.

The superfactorial of n is the product of the factorials of the numbers from 1 up to and including n: S(n) = 1! × 2! × 3! × … × n!.

For example, S(5) = 1! 2! 3! 4! 5! = 1 × 2 × 6 × 24 × 120 = 34560.

Here are three examples of where superfactorial pops up.

Vandermonde determinant If V is the n by n matrix whose ij entry is ij-1 then its determinant is S(n-1).

For instance, V is an example of a Vandermonde matrix.

Permutation tensor One way to define the permutation symbol uses superfactorial: Barnes G-function The Barnes G-function extends superfactorial to the complex plane analogously to how the gamma function extends factorial.

For positive integers n, Here’s plot of G(x) produced by Plot[BarnesG[x], {x, -2, 4}] in Mathematica.

More posts related to factorial Any number can start a factorial Alternating sums of factorials Defining zero factorial Variations on factorial How to compute log factorial.