# Catalan numbers

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

With C_0=1, the convolution recurrence and two exact closed forms define the same zero-based Catalan sequence.

mko-catalan-numbers · definition · v0.10.0



## Mathematical statement

C_(n+1)=sum over i=0..n of C_i*C_(n−i), and C_n=binomial(2n,n)/(n+1)=binomial(2n,n)−binomial(2n,n+1). Before division, certify that n+1 divides the central binomial coefficient; integer truncation is not a substitute. The domain is n≥0 and the C_0/C_1 indexing conventions must not be mixed.





## Representative formula (exact TeX)

C_n





## Assumptions


- n is a bounded nonnegative safe integer.
- Closed forms use exact BigInt arithmetic with a remainder-zero certificate.





## Symbol scope


- C_n — zero-based Catalan number





## Intuition, definition, and misconceptions

The base value is C_0=1.

The recurrence sums every split.

Closed forms need exact divisibility and cross-checks.

Do not begin at C_1 while using a C_0 formula, or silently truncate the quotient.





## Adopted derivation guide

[Catalan convolution and a divisibility certificate](https://ocme.evemisslab.com/en/derivations/derivation-mko-catalan-numbers-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Recurrence relation](https://ocme.evemisslab.com/en/mko/mko-recurrence-relation/) — Use the Catalan convolution.
- [Binomial theorem](https://ocme.evemisslab.com/en/mko/mko-binomial-theorem/) — The closed forms use binomial coefficients.
- [Basic counting principles](https://ocme.evemisslab.com/en/mko/mko-counting-principle/) — Finite combinatorial enumeration supports the sequence.
- [Natural-number divisibility](https://ocme.evemisslab.com/en/mko/mko-natural-divisibility/) — The closed-form quotient is integral.





## Verification boundary

Negative indices are outside the domain; this slice does not automatically prove every Catalan combinatorial bijection.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add analytic generating functions.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/catalan-numbers.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-catalan-numbers/index.md)