{
  "abstract": "Let $p>3$ be a prime, and let $b,c$ be integers with $p\\nmid b+2c$. We obtain an explicit congruence modulo $p^2$ for the sum of $\\binom{2k}{k}T_{2k}(b,c^2)/(4^k(k+1)(b+2c)^{2k})$ from $k=0$ to $(p-1)/2$, where $T_n(b,c)$ denotes the generalized central trinomial coefficient. This gives a parametric formula for the sum considered in a question of Wang and Cui. The proof uses a constant term representation and a logarithmic generating function to reduce the sum to two coefficients of an algebraic generating function. A differential identity and a coefficientwise polynomial congruence then yield the desired evaluation. The formula also applies when $p$ divides $b$ or $b-2c$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31423",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuqi Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c03wkqwjkwscq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31423",
  "fulltext_url": null,
  "id": "c03wkqwjkwscq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31423",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7215131815",
  "pdf_url": "https://arxiv.org/pdf/2609.31423",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c03wkqwjkwscq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.31423",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31423"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A parametric Catalan-type congruence for generalized central trinomial coefficients",
  "title_text": "A parametric Catalan-type congruence for generalized central trinomial coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/c03wkqwjkwscq",
  "venue": "arXiv",
  "versions": []
}