I'm considering some family of functions whose coefficients of their power series occur in the columns of the following matrix A (of course thought as of infinite size)
$ \qquad $ 
The second of that functions is $$ \small f_1(x) = - \sum_{k=1}^\infty \zeta(1-k) x^k \tag{1.1}$$ By evaluating it numerically for various $x$ using Noerlund sums I arrived at the guessed closed-form expression for it as $$ \small f_1(x) = \log(1/x) - \psi(1/x) \tag{1.2}$$
- Sanity-check: if I feed the following in W/A $$ \small \text{series [ log(t) - digamma(t) ]} $$ I get the same series, however in the form for some leading coefficients from which I conclude $$ \small [f_1(1/t) =] \sum_{k=1}^\infty -\zeta(1-k) \frac 1{t^k} \qquad \qquad \text{ expansion at } t=\infty$$ so it appears that the guessed closed-form is correct.
I have now the next function $$ \small f_2(x) = \sum_{k=2}^\infty a_{k,2} x^k \tag {2.1} $$ where the compositions of the $a_{k,2}$ is of $\zeta()$-values in a a bit more complicated form than the above, but with some describable pattern. However, on searching for a closed form for this I'm stuck.
To understand the pattern in the $a_{k,2}$ (and also that of $a_{k,3}$ in the fourth column and so on) I've found a very elegant solution, but I can't do the final step to get the closed forms. To describe that functions I involve Borel- or Laplace-transformation and -reversion.
The Borel-transformation of the series representation of $f_1(x)$ is $$ \small F_1(x) = - \sum_{k=1}^\infty \zeta(1-k) {x^k \over k! } \tag{3.1} $$ and this has a closed form $$ \small F_1(x) = \log( {\exp(x)-1\over x}) \tag{3.2}$$ Now the "elegant moment" is, that the Borel-transformation of the series representation of $f_2(x)$ $$ \small F_2(x) = \sum_{k=2}^\infty a_{k,2} \frac {x^k}{k!} \tag {3.3} $$ has simply the closed form $$ F_2(x) = (F_1(x))^2 \tag{3.3}$$ and moreover, for all following $c>2$ it is $$ F_c(x) = (F_1(x))^k \tag{3.4} $$
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 $k$ by the inverse Borel-transform. W/A helped for $F_1(x)$ but not for the $F_c(x)$ with higher column-index $c$.
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 $F_c(x)$ to $f_c(x)$ 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