  We investigate the coefficients of the polynomial \[
S_{m,r}^n(\ell)=r^n+(m+r)^n+(2m+r)^n+\cdots+((\ell-1)m+r)^n. \] We prove that
these can be given in terms of Stirling numbers of the first kind and
$r$-Whitney numbers of the second kind. Moreover, we prove a necessary and
sufficient condition for the integrity of these coefficients.
