We study the moments of orthogonal polynomial sequences (OPS) arising from tridiagonal matrices. We obtain combinatorial information about the sequence of moments of some OPS in terms of Motzkin and Dyck paths, and also in terms of the binomial transform. We then introduce an equivalence relation on the set of Dyck paths and some operations on them. We determine a formula for the cardinality of those equivalence classes, and use this information to obtain a combinatorial formula for the number of Dyck and Motzkin paths of a fixed length.