Wikipedia:Reference desk/Archives/Mathematics/2013 May 17

= May 17 =

Hyper-Exponential Function

 * $$f(x) = \frac1e \cdot \sum_{n=0}^{\infty}{n^x \over n!}$$

1. Why does this function return positive integer values for all positive integer arguments ?
 * Obviously, this is related to the fact that $$e = \sum_{n=0}^{\infty}{1 \over n!}$$ ... just not sure how exactly ...
 * $$f(n) \in \N^*\, \forall\ n \in \N^*$$ — Proof here

2. Are there any simpler methods of computing these positive integer values, either directly or recursively ?
 * {| class="wikitable" style="width: 400px; text-align: center"


 * 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)
 * 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)