Hankel determinants of linear combinations of consecutive Catalan-like numbers

Let (an)n≥0 be a sequence of the Catalan-like numbers. We evaluate Hankel determinants det[λai+j+μai+j+1]0≤i,j≤n and det[λai+j+1+μai+j+2]0≤i,j≤n for arbitrary coefficients λ and μ. Our results unify many known results of Hankel determinant evaluations for classic combinatorial counting coefficients, including the Catalan, Motzkin and Schröder numbers.