$\mathbb{N}$-algebraicity of zeta functions of sofic-Dyck shifts

We prove that the multivariate zeta function of a sofic-Dyck shift is the commutative series of some visibly pushdown language. As a consequence the zeta function of a sofic-Dyck shift is the generating function of a visibly pushdown language and is thus an $\mathbb{N}$-algebraic series.