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May 17

Hyper-Exponential Function

$f(x)=\frac{1}{e}\cdot \sum _{n=0}^{\mathrm{\infty }}\frac{n^x}{n!}$

1. Why does this function return positive integer values for all positive integer arguments ?

Obviously, this is related to the fact that $e=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}$ ... just not sure how exactly ...

$f(n)\in {\mathbb{N}}^{*},\mathrm{\forall }n\in {\mathbb{N}}^{*}$ — Proof here

2. Are there any simpler methods of computing these positive integer values, either directly or recursively ?

n	1	2	3	4	5	6	7	8	...
f(n)	1	2	5	15	52	203	877	4140	...

— Have you seen A000110, which gives hundreds of ways of computing these numbers?

70.162.4.242 (talk) 08:04, 17 May 2013 (UTC)

— See also

79.113.240.146 (talk) 15:00, 19 May 2013 (UTC)

3. Does this function have any special meaning or interesting properties ?

— They represent the number of partitions of a set with n members, as well as the coefficients of the Taylor series for $e^{e^x}$.

79.113.212.91 (talk) 07:52, 17 May 2013 (UTC)

— They're called Bell numbers and that formula above is Dobinski's formula.

Dmcq (talk) 12:32, 17 May 2013 (UTC)

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