  For any integer $m\geq 2$ and $r \in \{1,\dots, m\}$, let $f_n^{m,r}$ denote
the number of $n$-Dyck paths whose peak's heights are $im+r$ for some integer
$i$. We find the generating function of $f_n^{m,r}$ satisfies a simple
algebraic functional equation of degree $2$. The $r=m$ case is particularly
nice and we give a combinatorial proof. By using the Sulanke and Xin's
continued fraction method, we calculate the Hankel determinants for
$f_n^{m,r}$. The special case $r=m$ of our result solves a conjecture proposed
by Chien, Eu and Fu. We also enriched the class of eventually periodic Hankel
determinant sequences.
