  We characterize certain Riordan arrays by their $A$-matrices and $\rho$
sequences. We conjecture the form of a generic $A$-matrix which leads to Somos
$4$ sequences. We find an $A$-matrix that produces a Riordan quasi-involution,
and we study the $A$-matrices and $\rho$ sequences of the moment matrices of
certain perturbed orthogonal polynomials.
