We have stumbled upon the following finite alternating sum, which we have trouble analyzing. The sum is:

$$ S_n = \sum_{j=0}^n \frac{ (-1)^j e^{-j} }{j!} (n-j)^j $$

We have observed numerically that $S_n \approx 2 n e^{-n}$. We would like to establish whether this conjecture is true. More precisely, we would like to show that $S_n= \Theta(n e^{-n})$.

The sum is quite hard to evaluate numerically because the summands grow in absolute value initially and then decrease toward zero. The largest term is exponential in $n$ while the sum is conjectured to converge to zero exponentially fast. Any references or ideas are welcome!