For the opportunity of testing I include here just the original text of my question. Perhaps try to edit/update this "answer" to isolate the point from where the bug can be reproduced
I'm ölk considering some family of functions whose coefficients of their power series occur in the columns of the following matrix A
$ \qquad $ 
The second of that functions is
$$ f_1(x) = - \sum_{k=1}^\infty \zeta(1-k) x^k $$
By evaluating it numerically for various $x$ using Noerlund sums I arrived at the guessed closed-form expression for it as
$$ f_1(x) = \log(1/x) - \psi(1/x) $$
- Sanity-check: if I feed the following in W/A
$$ \text{series [ log(t) - digamma(t) ]} $$
I get the same series, however in the form for some leading coefficients from which I conclude
$$ [f_1(1/t) =] \quad \sum_{k=1}^\infty -\zeta(1-k) \frac 1{t^k} \qquad \qquad \text{ expansion at } t=\infty$$
so it appears that the guessed closed-form is correct.
I have now the next function
$$ f_2(x) = \sum_{k=2}^\infty a_{2,k} x^k $$
where the compositions of the $a_{2,k}$ is of $\zeta()$-values form the above, but with some pattern. On searching for a closed form for this I'm stuck.
To understand the pattern in the $a_{2,k}$ I've found a very elegant solution, but I can't do the final step. It involves Borel- or Laplace-transformation and -reversion.
The Borel-transformation of the series representation of $f_1(x)$ is
$$ F_1(x) = - \sum_{k=1}^\infty \zeta(1-k) {x^k \over k! } $$
and this has a closed form
$$ F_1(x) = \log( {\exp(x)-1\over x}) $$
Now the "elegant moment" is, that the Borel-transformation of the series representation of $f_2(x)$
$$ F_2(x) = \sum_{k=2}^\infty a_{2,k} \frac {x^k}{k!} $$
has simply the closed form
$$ F_2(x) = (F_1(x))^2 $$
and moreover, for all following $k>2$ it is
$$ F_k(x) = (F_1(x))^k $$
The problem is now that I don't know how to find from this the closed form for $f_2(x)$ (or $f_k(x)$ for higher $k$. W/A was of no help here.
Q1: What is the closed form for $f_2(x)$ and that of the following members of that family of functions?
Additional remark: I'm not experienced with Borel- or Laplace-transform. But while Pari/GP has a function
"serlaplace()" and this gives the expected transformation from $F_k(x)$ to $f_k(x)$ and I could experiment with this a bit, W/A gives another thing when called
"laplacetransform()" , actually it gave $f_1(x)/x$ instead $f_1(x)$. So question 2:
Q2: What is the correct expression for the series-forward&backward transformations: Borel? Laplace?
P.s.: I'm not good with tagging of questions like this. Please don't hesitate to improve the tagging if not optimal
