  Let $A$ be a proper Riordan array with general element $a_{n,k}$. We study
the one parameter family of matrices whose general elements are given by
$a_{2n+r, n+k+r}$. We show that each such matrix can be factored into a product
of a Riordan array and the original Riordan array $A$, thus exhibiting each
element of the family as a Riordan array. We find transition relations between
the elements of the family, and examples are given. Lagrange inversion is used
as a main tool in the proof of these results.
