A Generalization of the Ramanujan Polynomials and Plane Trees
Victor J. W. Guo, Jiang Zeng
Abstract
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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