Is it possible, solely with the function $f(x) = \sum_{n>0} a_nx^n$, to obtain the function $\sum_{n>0} \frac{a_n}{n!} x^n$?
as I proved in the own question:
$g(t) = \mathcal L^{-1} (\frac{f(\frac1s)}{s})$
where $\mathcal L^{-1}$ is the inverse of the Laplace transform.
If $f(s) = \frac1{1-s}$ then $g(t) = \mathcal L^{-1} (\frac1{s-1}) = e^t$ correct.
If $f(s) = 1 \implies g(t) = 1$ correct.
And so on