I'm reading an article where the authors derive the mass function of a compound distribution by considering the generating function. The generating function of interest for a random variable $N$ is a composition of two generating functions $G_Y(s)$ and $G_X(s)$, where \begin{equation}\tag{1} G_N(s)=G_Y(G_X(s))=e^{\lambda\left(\frac{(1-\rho)s}{1-\rho s}-1\right)}=e^{-\lambda} \sum_{m=0}^{\infty} \frac{1}{m !}\left(\lambda(1-\rho) s\right)^m\left(1-\rho s\right)^{-m} \end{equation} They then state,
Now, expand the probability generating function of $N$ in (1) and then, collecting the coefficient of $s^n$, we find an explicit expression for the probability mass function of N as $$ P(N=m)=e^{-\lambda} \sum_{i=0}^m \frac{1}{i !} {m-1\choose i-1}[\lambda(1-\rho)]^i \rho^{m-i}. $$
I don't see how this expression comes about from (1). I've tried using the negative binomial expansion identity of $$ (1-\rho)^{-r}=\sum_{k=0}^{\infty} {{k+r-1} \choose k}\rho^k $$ to substitute into (1), but I'm not getting anywhere. Any thoughts on what "expand the probability generating function" means or how to recover this PMF?