On a combinatorial problem of Asmus Schmidt

For any integer r ≥ 2, define a sequence of numbers {c^(r)_k} _k=0,1,…, independent of the parameter n , by ⁿ∑_k=0 ⁽ⁿ⁾_(k) ^{r (n+k)}_(k) ^r = ⁿ∑_k=0 ⁽ⁿ⁾_(k) ^(n+k)_(k)c^(r)_k, n = 0, 1, 2,... . We prove that all the numbers c^(r)_k are integers.