{
  "abstract": "For OEIS sequence A045406, the column-2 diagonal of the Lehmer-Comtet triangle A008296, R. J. Mathar contributed in September 2021 the conjectured order-2 P-recursive recurrence \\[ a(n) + (2n-7)\\,a(n-1) + (n-4)^{2}\\,a(n-2) \\;=\\; 0,\\qquad n \\ge 5. \\] We give a short proof. Detlefs's harmonic-number closed form $a(n) = (-1)^n (2 H_{n-3} - 3)(n-3)!$ for $n \\ge 3$ collapses the left-hand side, after factoring out $(-1)^n (n-5)! (n-4)$, to a polynomial identity in $n$ with coefficient $H_{n-4}$. The $H_{n-4}$-coefficient simplifies to $(n-3) - (2n-7) + (n-4) = 0$ (using $H_{n-3} = H_{n-4} + 1/(n-3)$ and $H_{n-5} = H_{n-4} - 1/(n-4)$); the constant remainder is $0$ for the same reason. The supplementary archive contains a SymPy script verifying both pieces symbolically, the e.g.f.\\ expansion against the harmonic closed form, and Mathar's recurrence numerically for $n = 5, \\ldots, 5000$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.12839",
  "author_total": 1,
  "authors": [
    {
      "name": "Tong Niu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dnr3ndgjqg73x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.12839",
  "fulltext_url": null,
  "id": "dnr3ndgjqg73x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.12839",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7161156336",
  "pdf_url": "https://arxiv.org/pdf/2605.12839",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dnr3ndgjqg73x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2605.12839",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.12839"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A short proof of Mathar's 2021 recurrence conjecture for the Lehmer-Comtet diagonal A045406",
  "title_text": "A short proof of Mathar's 2021 recurrence conjecture for the Lehmer-Comtet diagonal A045406",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/dnr3ndgjqg73x",
  "venue": "arXiv",
  "versions": []
}