How to estimate the following sum in terms of $n$?
$ \sum_{j_1,\ldots, j_{2k}\neq n}\frac{1}{(n-j_1)(n-j_2)\cdots(n-j_{2k-1})(n-j_{2k})}$ with $n+j_1, j_1-j_2, \ldots, j_{2k}-j_{2k-1}, n-j_{2k} \in \{-2,2\}$.
Do you have any ideas and how to proceed?
Thanks.