Generating Function for Odd Terms of Sequence

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

Consider the subsequence $\left\langle{b_n}\right\rangle := \left({a_1, a_3, a_5, \ldots}\right)$

Then the generating function for $\left\langle{b_n}\right\rangle$ is:


 * $\dfrac 1 2 \left({G \left({z}\right) - G \left({-z}\right)}\right)$

Proof
Hence the result.