A recent question asked about polynomial solutions to the functional equation $p(x^2)=p(x)p(x+1)$. Subsequently, Robert Israel posted an answer showing that solutions are necessarily of the form $p(x)=x^m(x-1)^m$.
What I had hoped to do myself was provide a solution by interpreting $p(x)$ as a generating function, with the functional equation leading to some kind of counting argument for the form of $p(x)$. Robert's answer complicates this, however, because $x^m(x-1)^m$ has coefficients of alternating signs. I then thought to route around this by replacing $x\to-x$ in the functional equation. But, while $p(-x)$ has positive coefficients, $p(-x+1)$ does not and so runs into the same issue.
That's as far as my thinking lead me. My question is: Is there some way to understand $p(x)$ (or some transformation of it) so as to provide a useful combinatorial interpretation of the functional equation?