On the divisibility of sums involving powers of multi-variable Schmidt polynomials

The multi-variable Schmidt polynomials are defined by $$S_n^{(r)}(x_0,\ldots,x_n):=\sum_{k=0}^n {n+k \choose 2k}^{r}{2k\choose k} x_k.$$ We prove that, for any positive integers $m$, $n$, $r$, and $\varepsilon=\pm 1$, all the coefficients in the polynomial $$\sum_{k=0}^{n-1}\varepsilon^k(2k+1) S_k^{(r)}(x_0,\ldots,x_k)^m$$ are multiples of $n$. This generalizes a recent result of Pan on the divisibility of sums of Ap\'ery polynomials.