On a combinatorial problem of Asmus Schmidt

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