{
  "id": "1611.01358",
  "version": "v1",
  "published": "2016-11-04T12:58:21.000Z",
  "updated": "2016-11-04T12:58:21.000Z",
  "title": "On some divisibility properties of binomial sums",
  "authors": [
    "Brian Y. Sun"
  ],
  "comment": "12 pages; to appear in IJNT; comments are welcome from everyone",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "In this paper, we consider two particular binomial sums \\begin{align*} \\sum_{k=0}^{n-1}(20k^2+8k+1){\\binom{2k}{k}}^5 (-4096)^{n-k-1} \\end{align*} and \\begin{align*} \\sum_{k=0}^{n-1}(120k^2+34k+3){\\binom{2k}{k}}^4\\binom{4k}{2k} 65536^{n-k-1}, \\end{align*} which are inspired by two series for $\\frac{1}{\\pi^2}$ obtained by Guillera. We consider their divisibility properties and prove that they are divisible by $2n^2 \\binom{2n}{n}^2$ for all integer $n\\geq 2$. These divisibility properties are stronger than those divisibility results found by He, who proved the above two sums are divisible by $2n \\binom{2n}{n}$ with the WZ-method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-04T12:58:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05",
      "11A07",
      "05A10",
      "11B65"
    ],
    "keywords": [
      "divisibility properties",
      "binomial sums",
      "divisibility results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}