Catalan numbers are a sequence of natural numbers that have many applications in combinatorial mathematics. They can be defined using a specific formula: C(n) = (2n)! / ((n + 1)!n!), where n is a non-negative integer. The first few Catalan numbers are 1, 1, 2, 5, 14, and 42. These numbers count various combinatorial structures, such as the number of ways to correctly match parentheses in expressions or the number of distinct binary search trees that can be formed with n nodes. Catalan numbers appear in diverse areas, including graph theory and geometry.