Regular "pentation"?

[BenStandeven]

[update:] after I've written the following I got aware, that this should be better threaded under something like "polynomial tetration/pentation" since the basic powerseries for the tetration as I used it here was derived from the "polynomial tetration" using the diagonalization of the truncated real (square) Bellmatrix and its diagonalization - *not* of the triangular Bellmatrix of the regular tetration

While usually I handle the tetration having a series which has the height-parameter in the exponents of its single terms, I played a bit around with a powerseries-representation, and especially a powerseries-epansion for b^^h for b=4 and beginning at x=0 (instead of x=1 as usual). So that powerseries gives b^^0 = 0, b^^1=1,b^^2=4 and so on. Obviously that powerseries has no constant term. It begins like

PHP Code:

b^^h= 0.938409102074*h - 0.167927302891*h^2 + 0.287554406742*h^3 - 0.151567019585*h^4 +... 


and for instance for b^^0.5 = 0.457214343478 .

That isn't going to give the standard version of pentation, then. You're iterating b^^(x-1) instead of b^^x.