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

For a real number $t$, let $r_{\mathrm{\ell }}(t)$ be the total weight of all $t$-large Schr"{o}der paths of length $\mathrm{\ell }$, and $s_{\mathrm{\ell }}(t)$ be the total weight of all $t$-small Schr"{o}der paths of length $\mathrm{\ell }$. 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.