A unified approach for the Hankel determinants of classical combinatorial numbers

We give a general formula for the determinants of a class of Hankel matrices which arise in combinatorics theory. We revisit and extend existent results on Hankel determinants involving the sum of consecutive Catalan, Motzkin and Schröder numbers and we prove a conjecture of [10] about the recurrence relations satisfied by the Hankel transform of linear combinations of Catalans numbers.