I have been wrestling with the following problem for some time now. If possible, I would prefer a hint rather than a full solution, because I would like to "solve" it myself.

Let $f(z)$ be a power-series and $[z^n]\{-\}$ denote the $n$'th coefficient. Show that the following holds, whenever $[z^0]{f(z)}=1$:

$$ \exp[\sum_{n,m>0}\sum_{j>0}j[z^{n+j}]\{f(z)^n\}[z^{m-j}]\{f(z)^m\}/(nm)] = \sum_{n\geq 0}[z^n]{f(z)^{(n-1)}}q^n\,. $$

Remark: This identity appears in comparing two generating series of the same geometric invariants computed using different methods.