{
  "id": "0910.3892",
  "version": "v7",
  "published": "2009-10-20T19:58:40.000Z",
  "updated": "2011-04-13T14:17:43.000Z",
  "title": "p-adic valuations of some sums of multinomial coefficients",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "16 pages",
  "journal": "Acta Arith. 148(2011), 63-76",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $m$ and $n>0$ be integers. Suppose that $p$ is a prime dividing $m-4$ but not dividing $m$. We show that $\\nu_p(\\sum_{k=0}^{n-1}\\frac{\\binom{2k}k}{m^k})$ and $\\nu_p(\\sum_{k=0}^{n-1}\\binom{n-1}{k}(-1)^k\\frac{\\binom{2k}k}{m^k})$ are at least $\\nu_p(n)$, where $\\nu_p(x)$ denotes the $p$-adic valuation of $x$. Furthermore, if $p>3$ then $$n^{-1}\\sum_{k=0}^{n-1}\\frac{\\bi{2k}k}{m^k}=\\frac{\\binom{2n-1}{n-1}}{4^{n-1}} (mod p^{\\nu_p(m-4)})$$ and $$n^{-1}\\sum_{k=0}^{n-1}\\binom{n-1}{k}(-1)^k\\frac{\\binom{2k}k}{m^k}=\\frac{C_{n-1}}{4^{n-1}} (mod p^{\\nu_p(m-4)}),$$ where $C_k$ denotes the Catalan number $\\binom{2k}{k}/(k+1)$. This implies several conjectures of Guo and Zeng [GZ]. We also raise two conjectures, and prove that $n>1$ is a prime if and only if $$\\sum_{k=0}^{n-1}multinomial{(n-1)k}{k,...,k}=0 (mod n),$$ where $multinomial{k_1+...+k_{n-1}}{k_1,...,k_{n-1}}$ denotes the multinomial coefficient $(k_1+...+k_{n-1})!/(k_1!... k_{n-1}!)$.",
  "revisions": [
    {
      "version": "v7",
      "updated": "2011-04-13T14:17:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "11A07",
      "05A10",
      "11S99"
    ],
    "keywords": [
      "multinomial coefficient",
      "p-adic valuations",
      "catalan number",
      "conjectures"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.3892S"
    }
  }
}