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As described in Analytic Combinatorics 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.