Generating Function for Odd Terms of Sequence

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

Consider the subsequence $\sequence {b_n} := \tuple {a_1, a_3, a_5, \ldots}$

Then the generating function for $\sequence {b_n}$ is:


 * $\dfrac 1 2 \paren {\map G z - \map G {-z} }$

Proof
Hence the result.