What is a Catalan number?

Catalan numbers form a sequence of natural numbers with many combinatorial applications. The nth Catalan number counts, among other things, the number of ways to fully parenthesize n+1 factors, the number of full binary trees with n+1 leaves, and the number of valid sequences of n pairs of parentheses.

What formula does this calculator use?

The closed-form formula is C(n) = (2n)! / ((n+1)! × n!). This is equivalent to the central binomial coefficient divided by (n+1): C(n) = C(2n, n) / (n+1).

Why is the limit set at n = 30?

C(30) is already 3,814,986,502,092,304 — a 16-digit number. Beyond n ≈ 30 standard 64-bit floating-point arithmetic loses precision. For larger values, arbitrary-precision libraries are needed.

C(0) = 1 by definition. This counts the single empty parenthesization, the single leaf-only binary tree, and so on.

How fast do Catalan numbers grow?

They grow exponentially: C(n) ≈ 4ⁿ / (n^(3/2) × √π). Each successive value is roughly 4× the previous one, which is why they become astronomically large quickly.

Catalan Number Calculator

Enter n to compute the nth Catalan number and see a table of the first N values in the sequence.

Compute Catalan Number

Enter n (0 – 30)

Catalan Sequence Table

Enter n and click Calculate to see the table.

Summary

The Catalan Number Calculator computes C(n) — the nth term of the Catalan sequence — using the closed-form formula C(n) = (2n)! / ((n+1)! × n!). It also displays a table of all Catalan numbers from C(0) through C(n), making it easy to study the full sequence at a glance. Catalan numbers appear throughout combinatorics: they count valid bracket sequences, full binary trees, monotonic lattice paths, and dozens of other combinatorial structures.

How it works

Enter a value of n (0 through 30) in the input field.
The calculator applies the closed-form formula C(n) = (2n)! / ((n+1)! × n!).
The result for C(n) is displayed immediately.
A table below lists every Catalan number from C(0) up to C(n).
Use the Copy button to copy the current result to your clipboard.

Use cases

Verify Catalan number values for combinatorics homework or research.
Count the number of valid parenthesizations of n+1 factors.
Find the number of distinct binary trees with n internal nodes.
Enumerate monotonic lattice paths that stay below the diagonal.
Cross-check results in computational geometry and graph theory problems.
Explore the growth rate of the Catalan sequence.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu