{"uid":"cap_5dD5V9baWcwBaT17pdsSP","slug":"catalan-number-calculator-b7ecdbb1","name":"Catalan Number Calculator","description":"The n-th Catalan number Cₙ = C(2n, n)/(n+1), counting balanced bracketings, binary trees, and many other structures. Requires n ≤ 10000.","url":"https://combinatorics.openverbs.com/v1/catalan","method":"POST","headers":{},"bodySchema":{"type":"object","$schema":"https://json-schema.org/draft/2020-12/schema","required":["input"],"properties":{"input":{"type":"object","required":["type","method","bodyType","body"],"properties":{"body":{"type":"object","required":["n"],"properties":{"n":{"type":"string","pattern":"^\\d+$","description":"Non-negative integer ≤ 10000."}},"additionalProperties":false},"type":{"type":"string","const":"http"},"method":{"enum":["POST"],"type":"string"},"bodyType":{"enum":["json","form-data","text"],"type":"string"}},"additionalProperties":false}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.004","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.004/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.004","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.004","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_J9rMH9DDdWLbR4PvmL2CA","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.004","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Computes the exact n-th Catalan number Cₙ = C(2n,n)/(n+1) for any n up to 10,000","exampleAgentPrompt":"What is the 50th Catalan number? I need the exact value for counting the number of valid balanced bracket sequences with 50 pairs.","exampleUseCases":[{"title":"Counting balanced bracket sequences","prompt":"How many valid ways can I arrange 12 pairs of parentheses so they're balanced? Give me the exact Catalan number C₁₂."},{"title":"Enumerating full binary tree shapes","prompt":"I'm studying binary tree structures — how many distinct full binary trees have exactly 8 internal nodes? That should be the 8th Catalan number."},{"title":"Polygon triangulation counting","prompt":"In how many distinct ways can a convex polygon with 15 sides be triangulated? Calculate the Catalan number for n=13 to figure this out."}],"resultDescription":"Returns the exact value of the n-th Catalan number as a (potentially very large) decimal integer string, along with its digit count for large values of n.","failureModes":["n exceeds 10000 — request rejected with validation error","n is negative or non-integer — schema validation failure","n is provided as a non-numeric string — pattern mismatch error","Network timeout for very large n due to computation of extremely large integers","Payment failure via x402 protocol resulting in 402 response"],"whenToPreferThis":"Choose this endpoint when you need the exact, arbitrary-precision Catalan number for a specific index n up to 10,000 — especially when the result is too large for standard 64-bit integers. Prefer this over general-purpose math libraries when you need results for large n without implementing big-integer arithmetic yourself.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-09-15T00:40:00.642Z","isFirstParty":false}