Catalan Number Calculator

Work out catalan number instantly with clear inputs, formula shown and shareable results.

About Catalan Number Calculator

The Catalan numbers Cₙ = C(2n, n)/(n + 1) count balanced bracket strings, binary trees with n internal nodes, triangulations of a convex polygon and monotonic lattice paths staying below the diagonal.

Formula

Catalan number

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

Worked example

With the default values below, Catalan Number Calculator returns nth Catalan number of 1,430. Change any input above and every figure here is recalculated in your browser.

Inputs used

Index naccepts 0–30: 8
How many terms to listaccepts 1–15: 8

Results

nth Catalan number: 1,430
Next Catalan number: 4,862
Central binomial C(2n, n): 12,870
First terms of the sequence: 1, 1, 2, 5, 14, 42, 132, 429

Step by step

1Values used
2Catalan number
3nth Catalan number= 1,430
4Next Catalan number= 4,862
5Central binomial C(2n, n)= 12,870
6First terms of the sequence= 1, 1, 2, 5, 14, 42, 132, 429

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is C₈?

C(16, 8)/9 = 12870/9 = 1430.

Why do so many problems give Catalan numbers?

They all reduce to the same recursive splitting structure Cₙ₊₁ = ΣCₖCₙ₋ₖ, which uniquely determines the sequence.