Periodic Column Partial Sums in the Riordan Array of a Polynomial

When $p(t)$ is a polynomial of degree $d$, $k$-th column of the Riordan array $(1\mathrm{/}(1-t^{d+1}),tp(t))$ is an eventually periodic sequence with the repeating part beginning at the $1+(k-1)(d+1)$-st term. The pre-periodic terms add up to the $(k-1)(d+1)$-st partial sum of the corresponding formal power series, and thus the Riordan array of $p(t)$ generates a sequence of column partial sums. We classify linear and quadratic polynomials, and present a particular family of polynomials of higher degrees, for which such sequences of column partial sums are eventually periodic.