{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-catalan-numbers",
  "semantic_family_id": "sf-catalan-numbers",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Catalan數"
  },
  "summary": {
    "zh-Hant": "Catalan數採C_0=1，並以convolution recurrence或兩個exact closed forms定義同一zero-based sequence。"
  },
  "statement": {
    "zh-Hant": "C_(n+1)=sum_{i=0}^n C_i C_(n-i)；C_n=binomial(2n,n)/(n+1)=binomial(2n,n)-binomial(2n,n+1)。division前必須驗(n+1)整除binomial(2n,n)，不可依integer truncation。domain n≥0且不可混用C_0/C_1 index convention。"
  },
  "formula": {
    "tex": "C_n",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mi>C</mi><mi>n</mi></msub></math>",
    "semantic_ast": {
      "type": "subscript",
      "base": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "subscript": {
        "type": "symbol",
        "name": "n",
        "glyph": "n"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "cfbf41dac773cbfcaed4836178f37ac893015cf1990a3b363d0ef5997973c242"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "n為bounded nonnegative safe integer。"
    },
    {
      "id": "a2",
      "zh-Hant": "closed-form以BigInt exact arithmetic及remainder-zero certificate。"
    }
  ],
  "symbols": [
    {
      "token": "C_n",
      "role": "zero_based_Catalan_number",
      "role_zh": "zero-based Catalan數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-recurrence-relation",
      "title_zh": "遞迴關係",
      "reason_zh": "Catalan convolution。"
    },
    {
      "id": "mko-binomial-theorem",
      "title_zh": "二項式定理",
      "reason_zh": "closed form coefficients。"
    },
    {
      "id": "mko-counting-principle",
      "title_zh": "基本計數原理",
      "reason_zh": "combinatorial enumeration。"
    },
    {
      "id": "mko-natural-divisibility",
      "title_zh": "自然數整除關係",
      "reason_zh": "closed-form integrality。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "base C_0=1。",
      "recurrence sums all splits。",
      "closed forms需exact divisibility與cross-check。"
    ],
    "common_misconception_zh": "從C_1起算卻沿用C_0公式，或用truncated division。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "負index不在domain；本輪不自動宣稱所有Catalan combinatorial bijections。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入analytic generating function。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-catalan-numbers.md 整理；2026-10-14 脈衝 R05（使用者手動接續 2026-11-11 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}