Implicit Divided Differences, Little Schr\"oder Numbers, and Catalan Numbers

Under general conditions, the equation $g(x,y)=0$ implicitly defines $y$ locally as a function of $x$. In this short note we study the combinatorial structure underlying a recently discovered formula for the divided differences of $y$ expressed in terms of bivariate divided differences of $g$, by analyzing the number of terms $a_n$ in this formula. The main result describes six equivalent characterizations of the sequence $\{a_n\}$.