Assuming I have a formal power series $$f = \sum_{n\geq 1} a_n X^n, \qquad (a_n \in \mathbb{Z}_{\geq 1}).$$I want to construct a new formal power series $$g = \sum_{n\geq 1} a_{a_n} X^n.$$
Question: Is there some kind of explicit description of an operator $T: \mathbb{Z}_{\geq 1}[[X]] \rightarrow \mathbb{Z}_{\geq 1}[[X]]$, such that $T(f) = g$? I know that above definition is "explicit", but I wonder if there another (maybe kind of analytic?) description. In particular, I am interested in the case when $f$ is given as a rational function, e.g. $f = \frac{X}{1-X}$ in which case $T(f) = f$.