I've proposed two sequences in the oeis: the pentational factorial and the tetrational factorial sequence (which unfortunately was rejected :( ).

I have constucted them starting from the classic factorial sequence: n!=n(n-1)!; then defined an exponential version n^!= n^(n-1)^! and finally used a tetrated and pentated version: n[4]!= n[4]((n-1)[4]!) n[5]!= n[5](n-1)[5]!.

I then thought of generalizing the concepts above.

In general, for any n-th hyperoperation we have:

a[n]b= b+1 for n<1 or a[n]b= a[n-1]a[n](b-1) otherwise.

Or equivalently: a[n]b= b+1 for n<1, a[1]b = a+b or a[n]b = a[n-1]a...[n-1]a with a occurring b times otherwise.

Define the k-torial of n for natural numbers k and n to be:

-1[k]! = -1 for k>3 or k<1;

0[2]! = 0! = 1;

0[k]! = 0 for k<2 or k>2;

n[k]! = n[k]((n-1)[k]!) otherwise.

The sequence n[k]! for nonpositive k is the sequence of all nonnegative integers. n[1]! are the triangular numbers. n[2]! is the factorial. n[3]! is exponential factorial or expofactorial or exponentorial. n[4]! the tetrational factorial or tetratorial. n[5]! the pentational factorial or pentatorial etc.

This sequence grows so quickly that a(4) cannot fit in the data section. This obviously happens even with many other sequences of the Torial family. Here are some that aren't included in the OEIS:

-the tetratorial sequence n[4]! where the terms are: -1, 0, 1, 2, 27, 4[4]27...

Sequences n[k]! for k>6 grow too quickly because only the same first 4 terms are visible;

-the operatorial or operational factorial sequence defined as n¡=n[n]! produces the terms 0, 1, 1, 2, 9, 4[4]27...

(See A257229 on the OEIS)

So I am defining K-torial, Torial, operatorial and (maybe) tetratorial and pentatorial, hexatorial, etc.

What do you think? Has there been anyone else who used those definitions before?