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I would like to ask if anyone would help me to find a general formula for the sum of m members of the following series. \begin{equation} \sum_{n=1}^{m} n^n = 1^1+2^2+\cdots + m^m \end{equation}

Thank you in advance.

Ayman Hourieh
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    AFAIK, there is no known general formula for this sum. – Ayman Hourieh Aug 25 '19 at 10:52
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    Nice little rule of thumb: If OEIS has your sequence, but doesn't have a closed expression for it, chances are there are no closed form expressions. Of course, absence of proof is not a proof of absence. Proving that it cannot be done (if it indeed cannot be done) is, presumably, difficult. – Arthur Aug 25 '19 at 10:52
  • Note: in a 1946 article by F. Underwood, nice upper and lower bounds were given for this sum (see here: https://www.jstor.org/stable/2306261) – Klangen Aug 25 '19 at 10:59
  • Related on MO: https://mathoverflow.net/questions/191216/the-hypertriangular-function-of-n – Klangen Aug 25 '19 at 11:00

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