Catalan Number Calculator

Compute Catalan numbers Cₙ = C(2n,n)/(n+1). Explore the sequence 1,1,2,5,14,42,132,…, recursive definition, 200+ combinatorial interpretations (binary trees, Dyck paths, triangulations), asymptotic growth, and Pascal's Triangle connection.

Cₙ
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Formula value check —
Combinatorial interpretation —
Extended More scenarios, charts & detailed breakdown ▾
C_n
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Central binomial C(2n,n) —
Interpretations —
Professional Full parameters & maximum detail ▾

Value & Formulas

C_n —
All formula forms —
Asymptotic growth —

Connections & Interpretations

Pascal's Triangle link —
Key interpretations (6 of 200+) —

How to Use This Calculator

  1. Enter n to compute the nth Catalan number instantly.
  2. Use First N Catalans tab to display the sequence C₀ through C_{N-1}.
  3. Use Recursive tab to see the expansion Cₙ=ΣCᵢ×Cₙ₋₁₋ᵢ.
  4. Use Professional for asymptotic growth, Pascal's link, and combinatorial interpretations.

Formula

Cₙ = C(2n,n) / (n+1) = (2n)! / ((n+1)! × n!)

Recursion: C₀=1, Cₙ = Σᵢ₌₀ⁿ⁻¹ Cᵢ × Cₙ₋₁₋ᵢ

Example

C₅ = C(10,5)/6 = 252/6 = 42. Counts: 42 ways to triangulate a heptagon (7-gon), 42 full binary trees with 5 internal nodes, 42 Dyck paths of length 10.

Frequently Asked Questions

  • The nth Catalan number is Cₙ = C(2n,n)/(n+1) = (2n)!/((n+1)!n!). The sequence begins 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, … It counts over 200 different combinatorial structures.
  • Cₙ counts: (1) binary trees with n internal nodes, (2) ways to parenthesize n+1 factors, (3) Dyck paths of length 2n (paths from (0,0) to (2n,0) using ±1 steps never going below 0), (4) triangulations of a convex (n+2)-gon, (5) non-crossing partitions of {1,…,n}.
  • C₀=1 and Cₙ = Σᵢ₌₀ⁿ⁻¹ Cᵢ × Cₙ₋₁₋ᵢ for n≥1. This counts by splitting a binary tree at the root: left subtree has i nodes, right subtree has n−1−i nodes.
  • Asymptotically Cₙ ~ 4ⁿ / (n^(3/2)√π). They grow roughly as 4ⁿ, so each term is approximately 4 times the previous one for large n.
  • Cₙ = C(2n,n) − C(2n,n+1), the difference between two adjacent central entries. Equivalently, Cₙ = C(2n,n)/(n+1), the central binomial coefficient divided by n+1.

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Sources & References (5) ▾
  1. Catalan Numbers — Richard P. Stanley — Cambridge University Press
  2. Catalan Number — Wolfram MathWorld — Wolfram Research
  3. OEIS A000108 — Catalan Numbers — OEIS Foundation
  4. An Introduction to the Theory of Numbers — Hardy & Wright — Oxford University Press
  5. A Course in Enumeration — Martin Aigner — Springer