Hankel determinants of sums of consecutive weighted Schröder numbers

We consider weighted large and small Schröder paths with up steps (1,1), down steps (1,-1) assigned the weight of 1 and with level steps (2,0) assigned the weight of t, where t is a real number. The weight of a path is the product of the weights of all its steps. Let and be the total weight of all weighted large and small Schröder paths from (0,0) to (2ℓ,0), respectively. For constants α, β, we derive the generating functions and the explicit formulae for the determinants of the Hankel matrices , , and combinatorially via suitable lattice path models.