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I'm looking for papers or recent results on the hypertriangular function of $n$:

$$H_t(n)= \displaystyle\sum\limits_{k=1}^{n} k^k$$

This is A001923 in the OEIS.

I don't have much experience with sums like this, but what makes finding a closed form so much more difficult that similar-looking sums such as the ones that appear in the "Sophomore's dream" identities?

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    $\begingroup$ A minor comment: in the Sophomore's dream identities one has series over $k^{-k}$ and not finite sums with general term $k^{k}$. The presence of a series allows you to relate the forms $x^{x}$ and $x^{-x}$ to the Taylor series of an exponential function (look at the first proof in the wikipedia page from the bottom to the top). In the OP one has simply a polynomial, instead. $\endgroup$
    – Avitus
    Commented Dec 22, 2014 at 0:04

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