Generalized Catalan Numbers And Catalan-like Numbers

The Catalan numbers form a very important sequence of natural numbers that occur in various counting problems in combinatorial mathematics. People mainly study their properties, including the Catalan numbers integral representation, the recurrence relation, the continued fraction, the lattice path representation, the Hankel matrix and the corre-sponding Catalan matrix. In addition, Motzkin numbers, large Schroder numbers, small Schroder numbers etc. are also very important in combinatorial mathematics and they have some similar properties with the Catalan numbers. Therefore, the research on Cata-lan numbers with its related sequences is the main content in combinatorial mathematics. In this paper, we present the combinatorial properties of generalized Catalan numbers and the new Taylor expansions for some special Catalan-like numbers respectively.This paper mainly consists of three chapters.In chapter 1, we give some definitions and terms, the state of development at home and abroad, and the organization of the paper.In chapter 2, we study the combinatorial properties of generalized Catalan numbers, including the recurrence relation, the continued fraction of the generating function, the log-convexity, the Hankel transform, the Hankel decomposition, and the total positivity of the corresponding Hankel matrix.In chapter 3, we give the new expansions for the generating function of Catalan-like numbers,Note that, unlike the usual Taylor expansions, the remainders in such expansions involve the generating function itself. In addition, we present the recurrence relations of the coefficients and bivariate generating functions of the remainders.