Generating Function of Sequence by Index

Theorem
Let $\map G z$ be the generating function for the sequence $\sequence {a_n}$.

Then:
 * $z \map {G'} z$ is the generating function for the sequence $\sequence {n a_n}$

where $\map {G'} z$ is the derivative of $\map G z$ $z$.

Proof
The result follows by definition of generating function.