Hankel determinants of sums of consecutive Motzkin numbers

We use combinatorial methods to evaluate Hankel determinants for the sequence of sums of consecutive t-Motzkin numbers. More specifically, we consider the following determinant:detmi+j+rt+mi+j+r+1t0⩽i,j⩽n-1,where t   is a real number and mkt is the total weight of all paths from (0,0)(0,0) to (k,0)(k,0) that stay above the x-axis and use up and down steps of weight one and level steps of weight t.