Asymptotic for a binomial sum Is it possible to derive an asymptotic expression as $n \to \infty$ for the following sum:
$$S_n(a,x) = \sum_{k=1}^n \binom{n}{k} \frac{k!^2 x^k}{k \cdot (a)_k} $$
$(a)_k$ is the rising factorial (Pochhammer symbol).
Euler-Maclaurin seems way too complicated here, and I'm not sure what else to try.

Not an answer.
$$S_n(a,x) = \sum_{k=1}^n \binom{n}{k} \frac{(k!)^2 }{k \, (a)_k}x^k$$
$$\frac{\partial S_n(a,x)}{\partial x} = \sum_{k=1}^n \binom{n}{k} \frac{(k!)^2 }{ \, (a)_k}x^{k-1}$$
$$\frac{\partial S_n(1,x)}{\partial x}=\frac{e^{\frac{1}{x}}\, x^n\, \Gamma \left(n+1,\frac{1}{x}\right)-1}{x} \quad \implies \quad  S_n(1,x)=\color{red}{\large ???}$$
