Is there any general formula for the sequence $S_n=\sum_{k=1}^n k^k$?

I have faced this series several times. Could it be possible to come up with a formula for the following sequence?

$S_n= \sum_{k=1}^n k^k = 1^1 + 2^2 + 3^3 + \ldots + n^n$

If yes, then what is that and how it could be proved?

Note: I had come through this question which I have found almost near to this series but still could not come up.

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