On a Combinatorial Problem of Asmus Schmidt

Abstract

For any integer $r\ge2$, define a sequence of numbers $\{c_k^{(r)}\}_{k=0,1,\dots}$, independent of the parameter $n$, by $$ \sum_{k=0}^n{n\choose k}^r{n+k\choose k}^r =\sum_{k=0}^n{n\choose k}{n+k\choose k}c_k^{(r)}, \qquad n=0,1,2,\dots. $$ We prove that all the numbers $c_k^{(r)}$ are integers.