Tetra-series

[Gottfried]

The way that I got the coefficients is slightly different than your method. I did this:
Let \( \mathbf{B} \) be a matrix defined by \( B_{jk} = \frac{1}{j!} \text{spow}_k^{(j)}(1) \), and let
\(
\begin{tabular}{rl}
f(x)
& = \sum_{k=1}^\infty f_k (x - 1)^k \\
& = \sum_{k=1}^\infty g_k ({}^{k}x) \\
F &= (f_0, f_1, f_2, ...)^T \\
G &= (g_0, g_1, g_2, ...)^T \\
\end{tabular}
\)
then
\( \mathbf{B}.F = G \)
so I thought, if we know G (1, -1, 1, -1, ...), then
\( F = \mathbf{B}^{-1}G \)
and when the matrix size is even I get the first series, and when the matrix size is odd, I get the second series.

Ah, now I understand, B is the matrix which transforms the f- into g - coefficients, G is given and F is sought...
B is not triangular here: how do you get the correct entries for its inverse, btw?

But whatever: I use this idea too, frequently.
However in many instances I found in our context of exponentiation and especially iterated exponentiation, that the inverse of some matrix X represents highly divergent series, such that systematically results which are correct using the non-inverted matrix are not correct for the inverse problem using the (naive) inverse of X.
This is especially the case for some matrix X, whose triangular LR-factors have the form of a q-binomial matrix.
Such LR-factors occur by a square matrix X = x_{r,c} = base^(r*c) or X = x_{r,c} = base^(r*c)/r! or the like, and if X shall be inverted by inversion of its triangular factors.
Such matrices X occur for example in the interpolation which I called "exponential polynomial interpolation" for the T-tetration (or sexp)-Bell-matrices. I used that matrix X also in the example for the "false interpolation for logarithm"-discussion. (But I could not yet find a workaround for the occuring inconsistencies with the inverse)

Now I don't see the precise characteristics of your B-matrix so far; I've just to actually construct one and to look into it to be able to say more. Let's see...

Gottfried