Hyperfactorial array notation

Hyperfactorial array notation is a large number notation invented by Lawrence Hollom. .

This notation is based on obvious extensions of factorials. There arises an extension: if we can multiply first n numbers, why not exponentiate it, tetrate, pentate them?

At the very first, Hollom defines the following notation:

\(n!m = n\uparrow^{m}(n-1)\uparrow^{m}(n-2)\cdots n\uparrow^{m} 3 \uparrow^{m} 2\)

Then he defines his creative variant of linear array notation, connected in his previous extension. Formally it can be defined as follows:

Rule 1. Condition: first entry is not an array and greater than 1:

\(n![k,@] = (((\cdots (((n![k-1,@])![k-1,@])!\cdots ))\) (with n [k-1]'s)

Rule 2. Condition: last entry is 1:

\(n![@,1] = n![@]\)

Rule 3. Condition: otherwise:

\(n![1,1,\cdots,1,1,k,@] = n![1,1,\cdots,1,[1,1,\cdots,1,1,1,@],k-1,@]\)

If the active entry (that should be decreased by one) is an array itself, then we apply these rules to that array.