Is it possible to express a power series with squared coefficients as a function of the series without squared coefficients?

Question

Suppose I have two sums, $P(x)$ and $Q(x)$:

$$P(x)\equiv \sum_{n=0}^N a_n x^n$$

$$Q(x)\equiv \sum_{n=0}^N a_n^2 x^n$$

Is there a way to express $Q(x)$ as a function of $P(x)$?

Context: I have a nonlinear recurrence for $a_{n} \rightarrow a_{n+1}$ which includes 1st, 2nd, and 3rd powers of $a_n$. I'd like to use $P(x)$ as a generating function for this recurrence, but I'm not sure how to express the $a_n^2$ (or $a_n^3$) sums in terms of $P(x)$, or if that's even possible.

Answer 1

How about

$$Q(x)=\sum_{k=0}^n \left(\frac{P^{(k)}(0)}{k!}\right)^2 x^k$$