Catalan Number Calculator

How to use the Catalan Number Calculator

1. 1Enter n (0 to 1,000).
2. 2Pick the units or options from the menus if the Catalan number tool offers them.
3. 3Read the Catalan number in the large result box.
4. 4Look at the extra rows for related figures.
5. 5Press Copy result to keep the answer.

When you need the Catalan number

Computer science students meet Catalan numbers when counting binary search tree shapes or ways to bracket a product. Mathematicians use them as a standard example when learning to count by bijection.

For a polygon with n + 2 sides, Cₙ is the number of ways to cut it into triangles with non-crossing diagonals. A hexagon (n = 4) can be cut in 14 ways. The same number counts the ways to fully bracket a product of n + 1 factors.

How the Catalan number is worked out

The nth Catalan number is Cₙ = (2n)! ÷ ((n + 1)! × n!). It also follows Cₙ₊₁ = Cₙ × 2(2n + 1) ÷ (n + 2), starting with C₀ = 1. C₅ = 10! ÷ (6! × 5!) = 42, so there are 42 ways to write five pairs of balanced brackets. The sequence runs 1, 1, 2, 5, 14, 42, 132, 429, 1430.

Every answer on this page comes from that one rule, so you can check any result by hand with a calculator. The figures shown underneath, such as other units, are conversions of the same answer.

• Exact big-number answers
• Shows digit count
• Lists the first sixteen values

Mistakes to avoid with Catalan number

Catalan numbers start at C₀, so the 5th number in a list that begins with C₀ is C₄ = 14, not 42. Check which index your problem uses. Also note that the numbers grow roughly like 4ⁿ, so large n gives very long answers.

On phone and computer

On a phone, type into the boxes one after another and read the Catalan number underneath; Copy result puts it on the clipboard for a message or a note. On a computer the inputs sit on the left and the answer on the right, so you can change one figure and watch the Catalan number move. The catalan number calculator works the same on both.

Checking your Catalan number by hand

A good habit is to test the catalan number calculator with a case you already know. C₅ = 10! ÷ (6! × 5!) = 42, so there are 42 ways to write five pairs of balanced brackets. The sequence runs 1, 1, 2, 5, 14, 42, 132, 429, 1430.

If your own figures give a surprising answer, repeat the working yourself: The nth Catalan number is Cₙ = (2n)! ÷ ((n + 1)! × n!). It also follows Cₙ₊₁ = Cₙ × 2(2n + 1) ÷ (n + 2), starting with C₀ = 1. Small differences usually come from rounding, from mixing units, or from a typing slip in one of the inputs. Catalan numbers grow quickly and come from counting problems, so related tools help. The Factorial Calculator shows the large products behind the formula, the Permutation and Combination Calculator covers other counting rules, and the Probability Calculator turns counts into chances.

Features

• Index starts at zero
• The numbers grow about fourfold per step
• Same value counts many structures
• Clear messages when the Catalan number inputs make no sense
• Copy the result with one tap
• Private: nothing is uploaded

Frequently asked questions