As described in <A HREF="http://algo.inria.fr/flajolet/Publications/ch4567.pdf">Analytic Combinatorics</A> by Flajolet and Sedgewick, page 4, a pole $t_0$ of the generating function gives the exponential asymptotics $P_n\sim t_0^n$. In this case $t_0=x+\sqrt{x^2-1}$. The subexponential factor can also be obtained from the generating function, but that requires more work.