Hankel determinants for generating functions of Catalan-like numbers

Let R-n(x) = Sigma(n)(i=0)r(i)x(n-i) be the generating functions of Catalan-like numbers (r(n))(n >= 0). We show that the corresponding Hankel determinants det[Ri+j(x)](0 <= i,j <= n) satisfy a three-term recurrence relation. As applications, we express the Hankel determinants of some classical combinatorial counting coefficients, including the Catalan numbers, the central binomial coefficients, the large and little Schroder numbers, in a unified way.