We consider Riordan arrays $\bigl(1/(1-t^{d+1}), ~ tp(t)\bigr)$. These are infinite lower triangular matrices determined by the formal power series $1/(1-t^{d+1})$ and a polynomial $p(t)$ of degree $d$. Columns of such matrix are eventually periodic sequences with a period of $d + 1$, and circulant matrices are used to describe the long term behavior of such periodicity when the column's index grows indefinitely. We also discuss some combinatorially interesting sequences that appear through the corresponding A - and Z - sequences of such Riordan arrays.