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 $n \rightarrow \infty$.