A Generalization of the Ramanujan Polynomials and Plane Trees
نویسندگان
چکیده
Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has recently introduced a polynomial sequence Qn := Qn(x, y, z, t) defined by Q1 = 1, Qn+1 = [x+ nz + (y + t)(n + y∂y)]Qn. In this paper we prove Chapoton’s conjecture on the duality formula: Qn(x, y, z, t) = Qn(x+nz+ nt, y,−t,−z), and answer his question about the combinatorial interpretation of Qn. Actually we give combinatorial interpretations of these polynomials in terms of plane trees, half-mobile trees, and forests of plane trees. Our approach also leads to a general formula that unifies several known results for enumerating trees and plane trees.
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