{
  "id": "1801.08905",
  "version": "v1",
  "published": "2018-01-26T17:30:51.000Z",
  "updated": "2018-01-26T17:30:51.000Z",
  "title": "On Motzkin numbers and central trinomial coefficients",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "21 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "The Motzkin numbers $M_n=\\sum_{k=0}^n\\binom n{2k}\\binom{2k}k/(k+1)$ $(n=0,1,2,\\ldots)$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\\ldots)$ given by the constant term of $(1+x+x^{-1})^n$ have many combinatorial interpretations. In this paper we establish the following surprising arithmetic properties of them with $n$ any positive integer: $$\\frac2n\\sum_{k=1}^n(2k+1)M_k^2\\in\\mathbb Z,$$ $$\\frac{n^2(n^2-1)}6\\,\\bigg|\\,\\sum_{k=0}^{n-1}k(k+1)(8k+9)T_kT_{k+1},$$ and also $$\\sum_{k=0}^{n-1}(k+1)(k+2)(2k+3)M_k^23^{n-1-k}=n(n+1)(n+2)M_nM_{n-1}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-26T17:30:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A10",
      "05A19",
      "11A07",
      "11B75"
    ],
    "keywords": [
      "central trinomial coefficients",
      "motzkin numbers",
      "constant term",
      "combinatorial interpretations",
      "surprising arithmetic properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}