Polynomial equation

This is only a partial solution, but: there is exactly one polynomial $P$ for each positive degree, and two constant polynomials $P(x) = -1,2$ that satisfy the equation. So a total of $\fbox{2016}$ in all.

If we let $T_n(x)$ be the Chebyshev polynomial of degree $n$ such that $T_n(\cos \theta) = \cos(n\theta)$ , then $P_n(x) = 2 T_n(x/2)$ satisfies the equation (exercise). So that gives the construction for the polynomials that satisfy the equation, but I am not sure how to show that these are the only ones.