I want to determine the probability mass function of a random variable $G$. I found a numerical scheme that gives me numerical values of the $z$-transform $g(\cdot)$ of $G$, i.e.
$g(z) = \sum_{n=0}^\infty \mathbb{P}(G=k) z^k,$
for $z\in\mathbb{R}$, where $\mathbb{P}(G=k)$ denotes the probability that $G$ equals $k$.
I am looking for a way to approximate $\mathbb{P}(G=k)$, for $k=0,1,2,\ldots$. I have been looking through the literature for numerical inversion algorithms, and it appears that many methods rely on function evaluations of $g$ in complex points.
My question is thus: Is there an inversion algorithm for $z$-transforms without knowing the functional form of the transform, i.e. where numerical real values suffice?