Let $A(x)$ be the generating function for $\{a_{n}\}_{n=0}^{\infty}$.
For what sequence is $A(x^{2})$ a generating function?
Let's say we have another generating function $B(x)$ for $\{b_{n}\}_{n=0}^{\infty}$. Then for example for $A(x) + B(x)$ the sequence would be $\{a_{n} + b_{n}\}_{n=0}^{\infty}$ and for $A(x)B(x)$ it would be a Cauchy product. For $A(x^{2})$ though I have no idea how to even start thinking about it.
If $A(x) = \sum_{n=0}^\infty a_n x^n$ is the generating function of the sequence $\{a_{n}\}_{n=0}^{\infty}$ then $$ A(x^2) = \sum_{n=0}^\infty a_n x^{2n} = a_0 + 0x + a_1x^2 + 0x^3 + a_2x^4 + \cdots = \sum_{n=0}^\infty b_n x^n $$ is the generating function of the sequence $\{b_{n}\}_{n=0}^{\infty}$ defined as $$ b_n = \begin{cases} a_{n/2} & \text{if $n$ is even,} \\ 0 & \text{if $n$ is odd.} \\ \end{cases} $$