On Hankel determinants for Dyck paths with peaks avoiding multiple classes of heights

For any integer m≥2 and a set V⊂{1,…,m}, let (m,V) denote the union of congruence classes of the elements in V modulo m. We study the Hankel determinants for the number of Dyck paths with peaks avoiding the heights in the set (m,V). For any set V of even elements of an even modulo m, we give an explicit description of the sequence of Hankel determinants in terms of subsequences of arithmetic progression of integers. There are numerous instances for varied (m,V) with periodic sequences of Hankel determinants. We present a sufficient condition for the set (m,V) such that the sequence of Hankel determinants is periodic, including even and odd modulus m.