A Ramanujan Sequence that Refines the Cayley Formula for Trees

Abstract

A further combinatorial study of Entries 16 and 17 in chapter 3 of Ramanujan's second notebook [1] leads to a refinement of Cayley's formula for counting trees and generalizes the recent results derived by Shor [13], and Dumont and Ramamonjisoa [5]. Our approach also yields a new proof of Clarke's extension of Cayley's formula [4].