About how big is $n^0+(n-1)^1+\cdots+0^n$?

Asked Dec 01 '11 at 00:08

Active Dec 01 '11 at 00:48

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Asymptotics of $1^n + 2^{n-1} + 3^{n-2} +\cdots + (n-1)^2 + n^1$

About how big is is the sum $\sum_{k=0}^n k^{n-k}$? At the least, can we get an upper bound on it that isn't terrible? (I would consider $(n+1)n^n$, or anything not significantly smaller than it, to be terrible.)

edited Apr 13 '17 at 12:21

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asked Dec 01 '11 at 00:08

Harry Altman

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FWIW, http://oeis.org/A026898 – lhf Dec 01 '11 at 00:12
Wolfram Alpha says that the maximum of $x^{n-x}$ ocurrs at $x = e^{W(n e)-1}$ and so an upper bound for the sum is $(n+1) \exp(e^{W(n e)-1}+n W(n e)-2n)$ but this is not tight. – lhf Dec 01 '11 at 00:30
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Related: http://math.stackexchange.com/questions/77790/asymptotics-of-1n-2n-1-3n-2-cdots-n-12-n1 – Zev Chonoles Dec 01 '11 at 00:31
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@Zev, it's the same question. The sum here is $1 + H_{n-1}$, to use the notation there. – Yuval Filmus Dec 01 '11 at 00:38
@Yuval: Ah, I was thrown by the different indexing :) – Zev Chonoles Dec 01 '11 at 00:58
Oops, completely missed that. Sorry, I'll delete this! Edit: Once it lets me... – Harry Altman Dec 01 '11 at 00:58
@Harry: Don't worry, happens to everyone! And I don't think that deletion is necessary; someone might search for your formulation, and we'd want them to find the same info. This is also the general opinion on this meta.SO post. – Zev Chonoles Dec 01 '11 at 01:32

About how big is $n^0+(n-1)^1+\cdots+0^n$?

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