TITLE: Closed form formula for $\sum\limits_{k=1}^n k^k$
QUESTION [13 upvotes]: Is there a way of finding a formula for $\sum\limits_{k=1}^n k^k$? Maybe I'm missing something really obvious, but I've looked around a bit on the Internet and I haven't been able to find anything.
So, what I'm looking for is a formula in closed form to generate the sequence $1,5,32,288,3413,\dots$

REPLY [4 votes]: I would write the inequality as
$$
n^n\left( 1+\frac{1}{4(n-1)} \right) \le 1^1 + 2^2 + \cdots + n^n < n^n \left(1+\frac{2}{e(n-1)}\right)
$$
to better show the bounds.