Hankel determinants of sums of consecutive weighted Schr{o}der numbers

For a real number $t$, let $r_ell(t)$ be the total weight of all $t$-large Schr{o}der paths of length $ell$, and $s_ell(t)$ be the total weight of all $t$-small Schr{o}der paths of length $ell$. For constants $alpha, beta$, in this article we derive recurrence formulae for the determinats of the Hankel matrices $det_{1le i,jle n} (alpha r_{i+j-2}(t) +beta r_{i+j-1}(t))$, $det_{1le i,jle n} (alpha r_{i+j-1}(t) +beta r_{i+j}(t))$, $det_{1le i,jle n} (alpha s_{i+j-2}(t) +beta s_{i+j-1}(t))$, and $det_{1le i,jle n} (alpha s_{i+j-1}(t) +beta s_{i+j}(t))$ combinatorially via suitable lattice path models.