We know that $\sum_{k=0}^n a^k \frac{n^{\underline k}}{k!} = (1+a)^n$. Is there a known (preferably closed) form for $\sum_{k=0}^n a^k n^{\underline k}$?
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1My copy of Mathematica gives $e^{1/a} a^n \Gamma (n+1,\frac{1}{a})$, where $\Gamma(s, x)$ is the Upper Incomplete Gamma Function. Do you consider that closed form? – George V. Williams Jun 30 '13 at 23:58
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maple gave hypergeom$([1,-n],[],-a)$ where the $[1,-n]$ are apparently "numerators" and the empty $[]$ means no "denominators". I'm not familiar with that, but it may be just a re-write of your sum. – coffeemath Jul 01 '13 at 00:01
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Twice you wrote and I don't know: what does $,n^{\underline k};$ mean? – DonAntonio Jul 01 '13 at 01:39
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@DonAntonio it means $n(n-1)(n-2)\ldots (n-k+1)$. – Ted Jul 01 '13 at 01:41
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Well I'll be...! I had no idea. Thanks @Ted – DonAntonio Jul 01 '13 at 01:44
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1related: http://math.stackexchange.com/questions/435780/binomial-like-sum – User3910 Jul 04 '13 at 03:24