I'm trying to find an asymptotic expansion for the following sum:
$$\sum_{k=2}^{n}\frac{n!}{k\left(n-k\right)!}=\sum_{k=2}^{n}\left(\begin{array}{c} n\\ k \end{array}\right)\left(k-1\right)!$$
for large $n$. According to Maple, this sum is equivalent to:
$$\frac{n^{2}-n}{2}\,_{3}F_{1}\left(1,2,2-n;3;-1\right)$$
where $_{3}F_{1}$ is a special case of the generalized hypergeometric function. But on the internet I cannot find any asymptotic expansion of $_{3}F_{1}$, because most work focused on other cases, e.g. $_{2}F_{1}$. I think it could be easier to work directly on the original sum. But by searching in previous posts I cannot find the specific sum I'm interested in. Moreover, asymptotic approximations of $\left(\begin{array}{c} n\\ k \end{array}\right)$ do not look very useful since they do not work for $k\approx n$. Help would be very appreciated!
$$\sum_{k=2}^{n}\binom{n}{k}(k-1)! = \sum_{k=2}^{n}\binom{n}{k}\int_{0}^{+\infty}x^{k-1}e^{-x}\,dx = \int_{0}^{+\infty}\frac{(1+x)^n-nx-1}{x}e^{-x}\,dx $$ is expected to behave like $\int_{0}^{+\infty}x^{n-1}e^{-x}\,dx = (n-1)!$ plus a perturbation due to the fact that $\frac{(1+x)^n-nx-1}{x}$ does not behave like $x^{n-1}$ in a right neighbourhood of the origin. More accurate approximations can be derived from Laplace's method.