Triangular numbers are [1] analogue of a factorial in hierarchy of hyper operations (1-summation, 2-multiplication, 3 -exponentiation, 4-tetration etc) So:

Tr (n) = n+(n-1) + (n-2) + ....2+1

n!= n*(n-1)*(n-2)*...*2*1

I am looking for properties of [3] analogue of factorial, its extension to complex numbers and generating function. There finite number of such factorials for each n, as exponentiation in not commutative, with 2 border cases:

Smallest:

Exponential factorial : n^(n-1)^(n-2)^ ...^2^1

Biggest:

3-factorial ( my invention for name) = 2^3^4^......(n-1)^n

Notice 1 is EXCLUDED from factorial here as it will destroy the whole idea giving result 1 for all n.

I am also interested in [0] analogue for factorial, [i] analogue , [-n] analogue - no idea yet what it means, even. With n-tations over [3] - [4] factorial, it is simpler though the number of factorials increase rapidly, but at least it is clear what it means:

4-factorial = 2[4]3[4]4[4].............(n-1)[4]n

I wonder if 2 stays, or shall we start from 3-just by analogy:

[1] factorial includes 0 as it is summation, but not negative numbers

[2] factorial includes 1 as it is multiplication, but not 0

[3] factorial includes 2 as it is exponentiation, but not 1

Conjecture :

[4] factorial includes 3?? as it is tetration, but not 2???

[5] factorial includes 4?? as it is pentation, but not 3???

[0] factorial includes -1? etc..

Ivars

Tr (n) = n+(n-1) + (n-2) + ....2+1

n!= n*(n-1)*(n-2)*...*2*1

I am looking for properties of [3] analogue of factorial, its extension to complex numbers and generating function. There finite number of such factorials for each n, as exponentiation in not commutative, with 2 border cases:

Smallest:

Exponential factorial : n^(n-1)^(n-2)^ ...^2^1

Biggest:

3-factorial ( my invention for name) = 2^3^4^......(n-1)^n

Notice 1 is EXCLUDED from factorial here as it will destroy the whole idea giving result 1 for all n.

I am also interested in [0] analogue for factorial, [i] analogue , [-n] analogue - no idea yet what it means, even. With n-tations over [3] - [4] factorial, it is simpler though the number of factorials increase rapidly, but at least it is clear what it means:

4-factorial = 2[4]3[4]4[4].............(n-1)[4]n

I wonder if 2 stays, or shall we start from 3-just by analogy:

[1] factorial includes 0 as it is summation, but not negative numbers

[2] factorial includes 1 as it is multiplication, but not 0

[3] factorial includes 2 as it is exponentiation, but not 1

Conjecture :

[4] factorial includes 3?? as it is tetration, but not 2???

[5] factorial includes 4?? as it is pentation, but not 3???

[0] factorial includes -1? etc..

Ivars