Let $c<1$. Let $X$ be a random variable with distribution: $$\forall k\in\mathbb{N}:\Pr[X=k]=\frac{e^{-ck}\cdot (ck)^{k-1}}{k!}$$ In fact, $X$ is an r.v. describing the total progeny in a Poisson Branching Process, i.e. the total amount of individuals across all generations.
Can we compute the probability generating function of $X$, i.e. $$G_X(s)=\sum_{k=1}^\infty\frac{e^{-ck}\cdot (ck)^{k-1}}{k!}\cdot s^k$$ Is it something nice, or perhaps not?