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Catalan number generator

Generate the nth Catalan number exactly, with big-integer precision. The Catalan numbers Cₙ = (2n)! / ((n+1)! n!) count an astonishing range of structures: balanced parentheses, binary trees, polygon triangulations, Dyck paths and hundreds more. Also lists C₀ through Cₙ.

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Need Catalan number Cₙ, Number of digits, C₀ … Cₙ? The Catalan number generator derives it from n (0 to 500) in one step. For instance, with n (0 to 500) = 10 it returns Catalan number Cₙ = 16796, Number of digits = 5 and C₀ … Cₙ = 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796.

How to use it

  1. Enter your values: n (0 to 500).
  2. Read the result instantly: Catalan number Cₙ, Number of digits, C₀ … Cₙ.

Frequently asked questions

How does the Catalan number generator work?

It takes n (0 to 500) and derives Catalan number Cₙ, Number of digits and C₀ … Cₙ from them. The calculation is live as you type, so the result updates on every change.

Which values does the calculator ask for?

A single value: n (0 to 500). Nothing else is required — no account, no file upload.

What does a typical calculation look like?

With n (0 to 500) = 10, the calculator returns Catalan number Cₙ = 16796, Number of digits = 5 and C₀ … Cₙ = 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796. Those figures come from running this exact tool, so you can reproduce them by entering the same values.

How much does the result change with different inputs?

It moves a lot. Using n (0 to 500) = 20 instead, Number of digits goes from 5 to 10 — which is why it is worth testing a few scenarios rather than trusting a single figure.

What does it give for smaller values?

Scaled down to n (0 to 500) = 5, Number of digits comes out at 2. The relationship is worth checking at both ends before you rely on a single result.

When would I actually use this?

Finding the nth term without listing every one before it, summing a long run in one step, and recognising which family a sequence belongs to.

What is the most common mistake?

Starting the index at the wrong end. Whether the first term is a₀ or a₁ shifts every result by one position, and the two conventions are both common.

What is the difference between the Catalan number generator and the Triangular number calculator?

This one returns Catalan number Cₙ and Number of digits; the Triangular number calculator returns Triangular number. That is the whole difference — open the one whose figure you need.

Is there a tool for the next step?

Fibonacci Number Checker is the closest one after this: Test any number up to 60 digits with the exact 5n^2 +/- 4 identity, with its index, its neighbours in the sequence and the ratio compared with the golden ratio.

What else is worth having open alongside it?

Happy number calculator and Perfect Number Checker — they come up in the same task often enough to be worth a second tab.

Further reading

All guides →
ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.ExplainerWhat Is a Z-Score? Standard Deviations Above the MeanA z-score is z = (x − μ) / σ — how many standard deviations a value sits from the mean. Learn how to compute it, read the standard normal curve, and turn it into a percentile.ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.How-toHow to Find the Distance Between Two PointsUse the distance formula d = √((x₂−x₁)² + (y₂−y₁)²) to measure the straight-line distance between two points on a plane, with a worked example and the midpoint too.How-toHow to Calculate the Slope of a Line: Rise Over RunFind the slope of a line from two points with m = (y₂ − y₁) / (x₂ − x₁), and read what a positive, negative, zero, or undefined slope tells you.