C_n, or the n-th Catalan number, is a sequence of natural numbers that have significant applications in combinatorial mathematics. It is defined by the formula C_n = (2n)! / ((n + 1)!n!), where "!" denotes factorial. Catalan numbers count various combinatorial structures, such as the number of valid parentheses combinations or the number of ways to triangulate a polygon. The sequence starts with C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, and so on. Each number in the sequence can be used to solve problems in areas like graph theory, tree structures, and dynamic programming. Catalan numbers are named after the French mathematician Eugène Charles Catalan.