Riordan Arrays and Shifted Hankel Determinants of OEIS A005773

We study two Riordan arrays associated with the Motzkin-prefix sequence (\href{https://oeis.org/A005773}{OEIS A005773}). By connecting their production matrices through a simple boundary factorization, we obtain a period-three transformation on Motzkin endpoint distributions and exact formulas for paths by final height modulo~$3$. In addition, a unified factorization for the shifted Hankel matrices leads directly to explicit closed forms for the third and fourth Hankel determinants, ${\mathrm{\Delta }}_n^{(3)}$ and~${\mathrm{\Delta }}_n^{(4)}$.