On a combinatorial problem of Asmus Schmidt

For any integer $r\ge 2$, define a sequence of numbers $\{c_k^{(r)}{\}}_{k=0}^{\mathrm{\infty }}$, independent of the parameter $n$, by $$\sum _{k=0}^n{\left(\genfrac{}{}{0px}{}{n}{k}\right)}^r{\left(\genfrac{}{}{0px}{}{n+k}{k}\right)}^r=\sum _{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)\left(\genfrac{}{}{0px}{}{n+k}{k}\right)c_k^{(r)},n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}c\mathrm{.}$$ We prove that all the numbers $c_k^{(r)}$ are integers.