Closed form expression for summation of P(n, k)

Question

I was wondering if there is a closed form expression for the following:

\begin{equation} \sum\limits_{k=1}^d P(n,k) \cdot n^{n-k} \end{equation}

This expression came up while analyzing an algorithm. The analysis is rather lengthy, but if asked, I can provide that. Thanks for your help!

Answer 1

$$\sum_{k=1}^d \frac{n! }{(n-k)!}n^{n-k}=e^n \Gamma (n+1,n)-e^n n!\frac{ \Gamma (n-d,n)}{\Gamma(n-d)}-n^n$$ where appears the complete and incomplete gamma functions.