I am looking for closed forms, or at least a good approximation for

$$f(n) = \sum_{k=1}^{k=n} \genfrac\{\}{0pt}{}{n}{k}(n)_kk$$

I know that

$$\sum_{k=1}^{k=n} \genfrac\{\}{0pt}{}{n}{k}(n)_k = n^n$$

I have the intuition that $f(n)$ is bounded above by $n^{n+1}$ and approaches $n^{n+1}$ for large $n$ but I am not entirely sure and don't know how to form a proof (or anti-proof).

Sorry if this question is too basic for math overflow, I wasn't sure if it belonged here or elsewhere.