Opetopes are algebraic descriptions of shapes corresponding to compositions in higher dimensions. As such, they offer an approach to higher-dimensional algebraic structures, and in particular, to the definition of weak ω-categories, which was the original motivation for their introduction by Baez and Dolan. They are classically defined inductively (as free operads in Leinster's approach, or as zoom complexes in the formalism of Kock et al.), using abstract constructions making them difficult to manipulate with a computer. Here, we present a purely syntactic description of opetopes and opetopic sets as a sequent calculus. Our main result is that well-typed opetopes in our sense are in bijection with opetopes as defined in the more traditional approaches. We expect that the resulting structures can serve as natural foundations for mechanized tools based on opetopes.