{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R201",
  "content_hash": null,
  "slug": "dclp-claim-generating-function-recurrence",
  "type": "claim",
  "title": "A divisor sum gives an exact coefficient recurrence",
  "summary": "The logarithmic derivative of the classical product computes every q_n from earlier coefficients.",
  "relevance": "For Eventual decrease for distinct cycle lengths in random permutations, record dclp-claim-generating-function-recurrence (“A divisor sum gives an exact coefficient recurrence”) records a bound, answer, status fact, or structural consequence. The record states: The logarithmic derivative of the classical product computes every q_n from earlier coefficients.",
  "relevance_source": "recorded",
  "body": "Cycle-index enumeration gives\n\\[\nQ(x)=\\prod_{k\\geq1}\\left(1+\\frac{x^k}{k}\\right).\n\\]\nDefine\n\\[\nB_m=\\sum_{d\\mid m}\\frac{(-1)^{m/d-1}}{d^{m/d-1}}.\n\\]\nExpanding the logarithmic derivative factor by factor gives\n\\[\n\\frac{xQ'(x)}{Q(x)}=\\sum_{m\\geq1}B_mx^m.\n\\]\nCoefficient comparison therefore yields the exact recurrence\n\\[\nq_0=1,\\qquad nq_n=\\sum_{m=1}^nB_mq_{n-m}\\quad(n\\geq1).\n\\]\nAfter multiplication by \\(n!\\), these are the integer permutation counts in OEIS A007838.",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "universal",
    "statement": "all coefficients q_n for n at least 0"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://oeis.org/A007838",
      "locator": "Generating function and recurrence; D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica 21 (1972), 379-388, Theorem 1"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A007838",
    "locator": "Generating function and recurrence; D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica 21 (1972), 379-388, Theorem 1"
  },
  "models": [],
  "relations": [
    {
      "slug": "dclp-problem-eventual-strict-decrease",
      "title": "Does the distinct-cycle-length probability decrease after n=30?",
      "object_type": "problem",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "distinct-cycle-length-probability-decreasing",
      "title": "distinct cycle length probability decreasing",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}