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 $\det_{1\le i,j\le n} (\alpha r_{i+j-2}(t) +\beta r_{i+j-1}(t))$, $\det_{1\le i,j\le n} (\alpha r_{i+j-1}(t) +\beta r_{i+j}(t))$, $\det_{1\le i,j\le n} (\alpha s_{i+j-2}(t) +\beta s_{i+j-1}(t))$, and $\det_{1\le i,j\le n} (\alpha s_{i+j-1}(t) +\beta s_{i+j}(t))$ combinatorially via suitable lattice path models.