  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.
