Hankel transform of linear combination of three consecutive Catalan numbers

In this paper, we consider the Hankel determinants of the linear combination of three consecutive Catalan numbers, and then three consecutive shifted Catalan numbers. For their computing, we apply known methods based on the connections between continued fractions, orthogonal polynomials and moment-determinants. The properties of orthogonal polynomials enable us to evaluate the generating function of the corresponding sequence Hankel determinants in closed form.