Catalan Numbers Calculator
Calculate Catalan numbers with BigInt support for large values. Catalan numbers count many combinatorial structures including balanced parentheses and binary trees.
Enter a position n (0 to 500) to calculate C(n)
The Catalan numbers form a sequence of natural numbers that occur in many counting problems involving recursively defined structures. The nth Catalan number is given by the formula C(n) = (2n)! / ((n+1)! * n!), which can also be written as C(n) = C(2n, n) / (n+1) where C(2n, n) is the central binomial coefficient.
The Sequence: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900...
Recurrence Relation: Catalan numbers satisfy C(0) = 1 and C(n+1) = sum of C(i) * C(n-i) for i from 0 to n. This recursive structure reflects how many combinatorial objects can be decomposed into smaller parts.
Counting Applications: C(n) counts the number of different ways to completely parenthesize a product of n+1 factors, the number of full binary trees with n+1 leaves, the number of paths from (0,0) to (2n,0) using steps (1,1) and (1,-1) that never go below the x-axis (Dyck paths), and the number of non-crossing partitions of a set with n elements.
Algorithm: This calculator uses the direct formula with BigInt arithmetic to handle arbitrarily large Catalan numbers. C(500) has over 290 digits, demonstrating the rapid growth of this sequence.
What do Catalan numbers count?
Catalan numbers count many combinatorial structures: the number of ways to correctly match n pairs of parentheses, the number of different binary search trees with n keys, the number of paths from (0,0) to (2n,0) that don't go below the x-axis, and the number of ways to triangulate a convex polygon with n+2 sides.
Why do Catalan numbers grow so quickly?
Catalan numbers grow roughly like 4^n/(n*sqrt(pi*n)). This exponential growth reflects the vast number of valid structures possible. For example, C(20) is over 6 billion, showing how many ways exist to arrange 20 pairs of balanced parentheses.
What is the formula for Catalan numbers?
The nth Catalan number is C(n) = (2n)!/((n+1)!*n!) = C(2n,n)/(n+1), where C(2n,n) is the central binomial coefficient. There's also a recurrence: C(n+1) = sum of C(i)*C(n-i) for i from 0 to n, with C(0) = 1.
Where are Catalan numbers used in computer science?
Catalan numbers appear in counting binary search trees, parsing expressions, evaluating stack-sortable permutations, and analyzing recursive algorithms. They're fundamental in combinatorics courses and appear in interview questions about tree structures and dynamic programming.