Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.8

Integer Sequences, Functions of Slow Increase, and the Bell Numbers

Rafael Jakimczuk
División Matemática
Universidad Nacional de Luján
Buenos Aires


In this article we first prove a general theorem on integer sequences $ A_n$ such that the following asymptotic formula holds,

$\displaystyle \frac{A_{n}}{A_{n-1}}\sim C n^{\alpha} f(n)^{\beta},$

where $ f(x)$ is a function of slow increase, $ C>0$, $ \alpha>0$ and $ \beta$ is a real number.

We also obtain some results on the Bell numbers $ B_n$ using well-known formulae. We compare the Bell numbers with $ a^n$ $ (a>0)$ and $ (n!)^h$ $ (0<h\leq 1)$.

Finally, applying the general statements proved in the article we obtain the formula

$\displaystyle B_{n+1}\sim
e\ \left(B_n\right)^{1+\frac{1}{n}}.$

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(Concerned with sequence A000110.)

Received December 9 2010; revised version received March 28 2011; May 4 2011. Published in Journal of Integer Sequences, May 10 2011.

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