Lattice paths and the Geode

Abstract: Let $t_1,t_2,\dots$ be variables, and let $S$ be the formal power series in the variables $t_1, t_2,\dots$ satisfying $S=1+\sum_{i=1}^\infty t_n S^n.$ Let $S_1 =\sum_{n=1}^\infty t_n$. Wildberger and Rubine recently showed that there is a formal power series $G$ in the $t_i$, which they called the Geode, satisfying $S=1+GS_1$. In this paper we discuss some of the properties of the Geode and of the related series $H=G/S$, which satisfies $S=1/(1-HS_1)$. We show that \begin{equation*} G=\biggl(1-\sum_{n=1}^\infty t_n (1+S+S^2+\cdots+S^{n-1})\biggr)^{-1}, \end{equation*} and \begin{equation*} H=\biggl( 1-\sum_{n=2}^\infty t_n (S+S^2+\cdots+S^{n-1})\biggr)^{-1}, \end{equation*} and we give combinatorial interpretations of $G$ and $H$ in terms of lattice paths.