The n-th Catalan number is a sequence of natural numbers that have many applications in combinatorial mathematics. It is defined by the formula C_n = \frac1n+1 \binom2nn , where \binom2nn is a binomial coefficient. Catalan numbers count various combinatorial structures, such as the number of valid parentheses combinations and the number of rooted binary trees with n nodes. The sequence starts with C_0 = 1 , C_1 = 1 , C_2 = 2 , C_3 = 5 , and so on. These numbers arise in many areas of mathematics, including graph theory, geometry, and computer science. The n-th Catalan number provides insight into the structure and counting of various combinatorial objects.