I'd like to know the asymptotic behaviour as $N\to\infty$ of the following sum

$$ Z_N(x) := 2^{-N/2} \sum_{k=0}^{N/2} \frac{N!}{k! (N-2k)!} (N-1)^{-k} (\sqrt{2} x)^{N-2k} $$

in order to compute $p(x):=\lim_{N\to\infty} \frac{log(Z_N(x))}{N}$ with $x\geq0$ (I already know this limit exists) .

I found the lower bound $p(x)\geq\log(x)$, that by graphical simulations seems to be very good when $x$ is big enough.

Can you help me to compute $p(x)$?