I ran across this sum when I was attempting to evaluate this Mellin transform: $$ \int_0^{\infty} x^{s-1} \: e^x \left( e^{e^{-x} \cdot e^{e^{-x}}}- e^{e^{-x}} \right) \: dx = \Gamma(s) \sum_{n=1}^{\infty} \frac{1}{n^s} \sum_{k=1}^{n} \frac{k^{(n-k+1)}}{k! (n-k+1)!} $$
The sum in question is symmetrical, that is: $$ \sum_{k=1}^{n} \frac{k^{(n-k+1)}}{k! (n-k+1)!} = \sum_{k=1}^{n} \frac{(n-k+1)^k}{k! (n-k+1)!} $$
I understand a closed form for the finite sum is highly unlikely but was interested if anyone was aware of connections to other special functions.
Thanks.
Edit: Sorry, I messed up the exponential bracketing. For better visual parsing the expression between the parenthesis is $$ \exp (\exp (-x) \cdot \exp (\exp (-x)))-\exp (\exp (-x)) $$