A generalization of the Ramanujan polynomials and plane trees

Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has recently introduced a polynomial sequence Qn:=Qn(x,y,z,t)Qn:=Qn(x,y,z,t) defined byQ1=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)Qn(x,y,z,t)=Qn(x+nz+nt,y,−t,−z), and answer his question about the combinatorial interpretation of QnQn. Actually we give combinatorial interpretations of these polynomials in terms of plane trees, forests of 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.