The Generating Function of the Catalan Numbers and Lower Triangular Integer Matrices

In the paper, by the Faà di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices. 1. Motivation The Catalan numbers Cn for n ≥ 0 form a combinatorial sequence of natural numbers that occur in tree enumeration problems such as “In how many ways can a regular n-gon be divided into n− 2 triangles if different orientations are counted E-mail addresses: [email protected], [email protected], [email protected], [email protected]. 2010 Mathematics Subject Classification. Primary 05A15; Secondary 11B65, 11B75, 11B83, 15A09, 15B36, 34A05, 34A34, 40E99.