It looks to me like the ordinary generating function of $P_n(x)$ is $$\sum_{n=0}^\infty P_n(x) s^n = - \sum_{k=0}^\infty \dfrac{(2k+1)! s^k}{2^k \prod_{j=1}^{k+1} (j^2 x s - 1)}$$
It looks to me like the generating function of $P_n(x)$ is $$\sum_{n=0}^\infty P_n(x) s^n = - \sum_{k=0}^\infty \dfrac{(2k+1)! s^k}{2^k \prod_{j=1}^{k+1} (j^2 x s - 1)}$$