Sum of a series and tetration?

Question

Is there an equation that can represent the value of the sum of a series (a sigma) if a tetration takes place inside it?

For example: $$ \sum\limits_{K=1}\limits^{N} {}^2\!K. $$ (Here ${}^2\!K = K^K$.)

Thanks! (:

Answer 1

In this paper, it is described that $$\sum_{k = 1}^n k^k = 1 + \sum_{k = 2}^n | \prod_{1 \leq i < j \leq k} (r_i - r_j)^2 | $$ where $r_1, \ldots, r_k$ are the roots of $z^k - 1 = 0$.