Nth Catalan Number Generator

About this calculator

The Catalan numbers are one of the most recurring sequences in combinatorics, defined as C(n) = (2n)! / ((n+1)! × n!), equivalently the central binomial coefficient C(2n, n) divided by (n+1). Rather than computing large factorials directly, this calculator uses a fast iterative recurrence — each term is derived from the previous one — which avoids overflow-prone intermediate values and scales efficiently to larger n.

Catalan numbers count a surprising range of structures that all turn out to be equivalent: the number of ways to correctly match n pairs of balanced parentheses, the number of distinct binary trees (or ordered rooted trees) with n internal nodes, the number of ways to triangulate a convex polygon with n+2 sides, and the number of monotonic lattice paths that stay below the diagonal — making them a staple in algorithm analysis, discrete mathematics courses, and computer science interview problems. Enter a 0-indexed n to generate C(n).