What is a good way of summarizing when "generating function" approach is useful to an audience of practitioners?

I'm giving a talk on training neural networks ([paper](https://arxiv.org/abs/2206.11124)), and want to motivate the idea of generating functions to audience not exposed to this concept.

One example I was considering was this:
consider following matrix sequences

$$
\begin{array}
\ A^0,A^1,A^2,\ldots\\
B^0,B^1,B^2\ldots\\
\end{array}
$$

We can consider a new sequence $(A+B)^n$
$$\ (A+B)^0,(A+B)^1,(A+B)^2,\ldots$$

If original sequences had generating functions $G_A, G_B$, combined sequence generating function is this:

$$G_{A+B}=(I-x G_A B)^{-1} G_A$$

In this case, "generating function" approach replaces matrix powering with inversion, so it helps when inversion is easy (ie, $A+B$ is diagonal + rank1), and doesn't help if matrix powering is easy.