Hankel determinants of shifted Catalan-like numbers

Let (a0,a1,a2,…) be the sequence of Catalan-like numbers. We evaluate the Hankel determinants of the shifted sequence (0,a0,a1,a2,…). As an application, we settle Barry's three conjectures about Hankel determinant evaluations of certain sequences in a unified approach. We also provide some Somos sequences by means of Hankel determinants of the (shifted) Catalan-like numbers.