Decomposition of Triply Rooted Trees

An Identity Conjectured by Lacasse via the Tree Function

“…Then Lacasse [3] conjectured that ξ 2 (n) = ξ(n) + n. There are now three independent proofs [6,1,5]. Here, we want to shed additional light on the matter, by using the tree function (equivalent to Lambert's W -function [7]) and linking the enumeration to the celebrated Q-function of Ramanujan [2].…”

A Simple Proof of an Identity of Lacasse

“…Recently, by applying the Hurwitz identity on multivariate Abel polynomials, Younsi [7] gave an algebraic proof of this conjecture. Later, using a decomposition of triply rooted trees into three doubly rooted trees, Chen, Peng and Yang [1] gave it a nice combinatorial interpretation.…”