Journal of Integer Sequences, Vol. 15 (2012), Article 12.4.3

A Generalized Apéry Series


M. L. Glasser
Department of Physics
Clarkson University
Postdam, NY 13699-5820
USA

Abstract:

The inverse central binomial series

$\displaystyle S_k(z)=\sum_{n=1}^{\infty}\frac{n^k z^n}{\binom{2n}{n}},$    

popularized by Apéry and Lehmer, is evaluated for positive integers $ k$ along with the asymptotic behavior for large $ k$. We show that the value $ z=2$, as commented on by D. H. Lehmer, provides a unique relation to $ \pi$.

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(Concerned with sequences A008277 and A145557.)

Received October 8 2011; revised version received April 2 2012. Published in Journal of Integer Sequences, April 6 2012.

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