The Narayana polynomials (OEIS A001263) are the h-polynomials of the associahedra (the Stasheff polytopes) and their dual simplicial polytopes (cf. the Fomin and Reading ref in the OEIS entry).

Are these polynomials related to the Ehrhart series (cf. also Computing the Continuous Discretely by Beck and Robins) of any families of polytopes?

Are there any properties of the Narayana polynomials that preclude them from being the numerator polynomials associated to the Ehrhart series rational functions of any family of polytopes?