Product of Generating Functions/General Rule

Theorem
Let $G_0 \left({z}\right), G_1 \left({z}\right), G_2 \left({z}\right), \ldots$ be any number of generating functions (up to countably infinite) for the sequences $\left\langle{a_0 n}\right\rangle, \left\langle{a_1 n}\right\rangle, \left\langle{a_2 n}\right\rangle, \ldots$

Then: