Define a sequence $<x_n>$ by $x_1 = 1, x_2 = x, x_{n+2} = xx_{n+1} + nx_n, n \ge 1$. Consider the polynomial $P_n(x) = x_{n-1}x_{n+1} - x_n^2$, for each $n \ge 2$. Prove or disprove that the coefficients of $P_n(x)$ are all non-negative, except for the constant term when $n$ is odd.