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Conjecture about 3,4,5 ..etc factorials based on analogy - Ivars - 06/01/2008
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 |