The Catalan number is a sequence of natural numbers that have many applications in combinatorial mathematics. The nth Catalan number can be calculated using the formula C(n) = (2n)! / ((n + 1)!n!), where n is a non-negative integer. These numbers count various combinatorial structures, such as the number of ways to correctly match parentheses or the number of distinct binary search trees that can be formed with n nodes. Catalan numbers appear in various mathematical problems, including counting paths in a grid and determining the number of triangulations of a polygon. The sequence starts with C(0) = 1, C(1) = 1, C(2) = 2, C(3) = 5, and continues indefinitely. Their significance extends beyond pure mathematics, influencing fields like computer science and physics.