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In the paper, the authors review and survey some new results on the large Schröder numbers. These results were obtained by the authors and their coauthors from January 2016 and can be summarized up as follows. (a) By studying the generating function of the large Schröder numbers, the authors and their coauthors establish several explicit formulas and a recursive formula for the large Schröder numbers and present relations between the Schröder numbers and central Delannoy numbers. (b) By utilizing the Cauchy integral formula in the theory of complex functions, the authors and their coauthors build several integral representations for the large Schröder numbers and their generating function. (c) By employing integral representations for the large Schröder numbers, the authors and their coauthors find some properties, including the convexity, complete monotonicity, product inequalities, and determinantal inequalities, of the large Schröder numbers. 1. What are the Schröder Numbers? Let us recall some basic knowledge about the Schröder numbers. 1.1. What are the Schröder Numbers? In combinatorics and number theory, there are two kinds of Schröder numbers, the large Schröder numbers Sn and the

Explicit and recursive formulas, integral representations, and properties of the large Schröder numbers