LaTeX
C_n=\frac{1}{n+1}\binom{2n}{n}Variables
- n: nonnegative integer index
- C_n: nth Catalan number
How to use this formula
Counts many recursively structured objects such as balanced parentheses and polygon triangulations.
Important notes
- Equivalent forms include C_n=(2n)!/(n!(n+1)!).
- Different counting problems require a proof of the Catalan correspondence.
Quick example
C₄ = 14, so a convex hexagon has 14 triangulations.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Equivalent forms include C_n=(2n)!/(n!(n+1)!).
- For the Catalan Number, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- Define whether selections are ordered, repeated, labeled, or constrained before applying the counting formula.
Worked example
C₄ = 14, so a convex hexagon has 14 triangulations.
Common mistakes
- When copying Catalan Number, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Verify the result of Catalan Number with a known case, inverse operation, dimensional check, or independent calculation before publishing it.
Continue the workflow
Use Catalan Number in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
C_n=\frac{1}{n+1}\binom{2n}{n}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Catalan Number used for?
Counts many recursively structured objects such as balanced parentheses and polygon triangulations.
Can I copy this formula as LaTeX?
Yes. Copy C_n=\frac{1}{n+1}\binom{2n}{n} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.