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 *i*^{j−1} then its determinant is *S*(*n* − 1). For instance,

V is an example of a Vandermonde matrix.

## Latin squares

The number of Latin squares of size *n* is bounded below by *S*(*n*). More on Latin squares and upper and lower bounds on how many there are here.

## 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.

Your definition for G(n) seems to only be defined when n is a positive integer.

The identity for G above is only valid for positive integers. It’s not a definition. There is a closed-form definition of G. It’s a little complicated, but if I remember correctly it just involves gamma, psi (derivative of log gamma), and exp.