In a discrete mathematics past paper, I am asked to find the generating function for the sequence $$\langle a_0, 0, a_2, 0, a_4, 0, \ldots \rangle,$$ given that $A(x)$ is the generating function for the sequence $$\langle a_0, a_1, a_2, a_3, \ldots \rangle.$$
I have been struggling with this for a while, with no success.
I think that I am missing something simple and would appreciate a hint to help me solve the problem.
You want to kill the terms of odd degree, which can be done by projecting onto the space of even power series parallel to the space of odd power series (the two spaces are complementary inside the space of all power series). If your generating series is given by an expression $A(x)$ (and unless you are working over a field of characteristic$~2$, which I will suppose) you can build this "even image" of $A$ as a linear combination of $A(x)$ and of the result $A(-x)$ of substituting $-x$ for $x$; the coefficients are easily found.