Given a sequence ${a_k}, k=1,2,3...$, find the generating function for ${a_2k}$

Question

This is a problem on an exam review sheet for a discrete mathematics course.

Consider a generating function

$$F(x)=a_0+a_1x+a_2x^2+...$$

Using operations on generating functions, obtain a generating function for the sequence $a_0, a_2, a_4, a_6, ..., a_k$ .

I guess I'm just kind of confused as to what exactly I'm supposed to find. Would I need to find a way to subtract the odd power terms in $F(x)$, or do I find a function of the form

$$G(x)=a_0+a_2x+a_4x^2+a_6x^3+...+a_{2k}x^k?$$

If I already wrote out this function is it given or do I need to somehow put it in terms of $F(x)$? Sorry if this is a silly question, I'm sure if I just figure out what I'm supposed to be finding I can figure out the rest on my own.

Answer 1

$$G(x) = \frac12 \left [F \left ( \sqrt{x} \right ) + F \left (- \sqrt{x} \right ) \right ] $$

I feel the answer is self-explanatory. As an additional exercise, try computing the G.F. for the sequence $a_1, a_3,\ldots$.

Answer 2

Hint: We have $$ e^x=\cosh(x)+\sinh(x) $$ so $$ \cosh(x)=e^x-\sinh(x) $$