{
  "id": "1412.5415",
  "version": "v1",
  "published": "2014-12-10T00:38:55.000Z",
  "updated": "2014-12-10T00:38:55.000Z",
  "title": "Proof of some conjectures of Z.-W. Sun on the divisibility of certain double-sums",
  "authors": [
    "Victor J. W. Guo",
    "Ji-Cai Liu"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Z.-W. Sun introduced three kinds of numbers: \\begin{align*}S_n=\\sum_{k=0}^{n}{n\\choose k}^2{2k\\choose k}(2k+1),\\qquad s_n=\\sum_{k=0}^{n}{n\\choose k}^2{2k\\choose k}\\frac{1}{2k-1}, \\end{align*} and $S_n^{+}=\\sum_{k=0}^{n}{n\\choose k}^2{2k\\choose k}(2k+1)^2$. In this paper we mainly prove that \\begin{align*} 4\\sum_{k=0}^{n-1}kS_k\\equiv \\sum_{k=0}^{n-1}s_k\\equiv \\sum_{k=0}^{n-1}S_k^{+}\\equiv 0\\pmod{n^2}\\quad\\text{for $n\\geqslant 1$}, \\end{align*} by establishing some binomial coefficient identities, such as \\begin{align*} 4\\sum_{k=0}^{n-1}kS_k=n^2\\sum_{k=0}^{n-1}\\frac{1}{k+1}{2k\\choose k}(6k{n-1\\choose k}^2+{n-1\\choose k}{n-1\\choose k+1}). \\end{align*} This confirms several recent conjectures of Z.-W. Sun.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-10T00:38:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B65",
      "05A10"
    ],
    "keywords": [
      "conjectures",
      "divisibility",
      "double-sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.5415G"
    }
  }
}