Asymptotic expansion of a sum containing binomial coefficients 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!
Can be done in a similar way to this. Let $k = n - k$ for convenience:
$$S = \sum_{k = 2}^n \frac {n!} {k (n - k)!} =
\sum_{k = 0}^{n - 2} \frac {n!} {(n - k) k!} =
(n - 1)! \sum_{k = 0}^{n - 2} \frac 1 {k!} \frac 1 {1 - \frac k n}.$$
The main contribution comes from small values of $k$, and it is sufficient to use an approximation of the summand that holds for large $n$ and fixed $k$. The $i$th order term in the approximation will be simply $(k/n)^i$. Then the summation range in $S$ can be extended to infinity, and we get a complete asymptotic series for $S$:
$$S \approx e (n - 1)! \sum_{i = 0}^\infty \frac {B_i} {n^i},$$
where $B_0 = 1, B_1 = 1,  .\!.\!.$ are the Bell numbers.
You must be more specific. Do you want an asymptotic expansion for large $n$? if so, are you aware of Sterling formula to approximate $n!$ for large $n$? This should help you finding your answer.