A Grammatical Calculus for the Ramanujan Polynomials

As remarked by Berndt, no combinatorial perspective seems to be alluded in the original definition of the Ramanujan polynomials. On a different scene, a recursive algorithm to generate rooted trees has been devised independently by Shor and Dumont-Ramamonjisoa. Zeng discovered the connection between the Ramanujan polynomials and the enumeration of rooted trees by number of improper edges. We present a proper labeling scheme for rooted trees by employing an extra label. Harnessed by this grammar, we develop a calculus heavily depending on the constant properties for the Ramanujan polynomials. From the grammatical formulation, we recover the defining equation of Ramanujan on an implicit function. So the two themes of Ramanujan converge to one combinatorial structure. Moreover, we provide a grammatical treatment of a bijection behind the recursion independently due to Shor and Berndt-Evans-Wilson.