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Integer Sequences, Functions of Slow Increase
and the Bell Numbers Rafael Jakimczuk
División Matemática
Universidad Nacional de Luján
Buenos Aires
Argentina
mailto:jakimczu@mail.unlu.edu.arjakimczu@mail.unlu.edu.ar

In memory of my sister Fedra Marina Jakimczuk (1970-2010)

Abstract:

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

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

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

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