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Counting, Permutations & Combinations

Catalan Number Calculator

Calculate Catalan numbers for lattice paths, balanced structures, and related counting problems.

Counting & Combinatorics

Calculate a Catalan number

Exact Cₙ

Exact count

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Enter valid values to see the result.

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Quick guide

How to use this calculator

  1. Enter a non-negative index n.
  2. Read the exact Catalan number Cₙ.
  3. Apply it only when your problem matches a Catalan-counted structure.

Calculation method

Divide the central binomial coefficient

The nth Catalan number is C(2n,n)/(n+1).

Worked example

Fifth Catalan number

The fifth Catalan number is 42.

C₅ = C(10,5)/6 = 42

Supported inputs

Precision and limits

Discrete domain

Inputs are whole counts. Negative, fractional, grouped, and scientific-notation inputs are rejected.

Exact integers

Results use arbitrary-precision integers and are never rounded. Inputs and output size are bounded to keep the page responsive.

Interpretation

Catalan numbers count many different structures, but only when the structure satisfies the corresponding restrictions.