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Tetration is repeated exponentiation. So for example a[4]4 = a^^4 = a^a^a^a.
Pentation is repeated tetration. So for example a[5]3 = a^^^3 = a^^a^^a.
a[n](-1) = 0 and a[n]0 = 1 for every a and n>3. Let a^^^b be a^^a^^...^^a with a occurring b times.
In general, for any n-th hyperoperation we have:
a[n]b = b+1 for n<1 and a[n]b= a[n-1]a[n](b-1) otherwise.
Or equivalently, a[n]b = b+1 for n<1 and 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...
All of the above sequences have offset -1.
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