Generating Function of Multiple of Parameter

Theorem
Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$.

Let $c$ be a constant.

Then $G \left({c z}\right)$ be the generating function for the sequence $\left\langle{b_n}\right\rangle$ where:
 * $\forall n \in \Z_{\ge 0}: b_n = c^n a_n$

Proof
Hence the result by definition of generating function.