Sums involving moments of reciprocals of binomial coefficients

Belbachir, Hacène ; Rahmani, Mourad ; Sury, B.

Journal of integer sequences, Tome 14 (2011) no. 6

Résumé

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$.

Détail

Keywords: binomial coefficient, recurrence relation, generating function, asymptotic expansion