Journal of Integer Sequences, Vol. 14 (2011), Article 11.6.6

Sums Involving Moments of Reciprocals of Binomial Coefficients


Hacène Belbachir and Mourad Rahmani
University of Sciences and Technology Houari Boumediene
Faculty of Mathematics
P. O. Box 32
El Alia
Bab-Ezzouar 16111
Algiers
Algeria

B. Sury
Statistics & Mathematics Unit
Indian Statistical Institute
8th Mile Mysore Road
Bangalore 560059
India

Abstract:

We investigate sums of the form $ \sum_{0\leq k\leq n}k^{m}\binom{n}{k}^{-1}.$ We establish a recurrence relation and compute its ordinary generating function. As application we give the asymptotic expansion. The results extend the earlier works by various authors. In the last section, we establish that $ \sum_{0\leq k\leq n} \frac{k^{m}}{n^m} \binom{n}{k}^{-1}$ tends to $ 1$ as $ n \rightarrow \infty$ and that $ \sum_{0\leq k\leq n-m}k^{m}\binom{n}{k}^{-1}$ tends to $ m!$ as $ n \rightarrow \infty$.

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(Concerned with sequences A008277 A008292 A028246.)

Received December 1 2010; revised version received December 8 2010; May 28 2011. Published in Journal of Integer Sequences, June 10 2011.

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