{
  "id": "1609.06810",
  "version": "v1",
  "published": "2016-09-22T03:57:57.000Z",
  "updated": "2016-09-22T03:57:57.000Z",
  "title": "Hankel-type determinants for some combinatorial sequences",
  "authors": [
    "Bao-Xuan Zhu",
    "Zhi-Wei Sun"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper we confirm several conjectures of Z.-W. Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Ap\\'ery numbers. For any nonnegative integer $n$, define \\begin{gather*}f_n:=\\sum_{k=0}^n\\binom nk^3,\\ D_n:=\\sum_{k=0}^n\\binom nk^2\\binom{2k}k\\binom{2(n-k)}{n-k}, \\\\b_n:=\\sum_{k=0}^n\\binom nk^2\\binom{n+k}k,\\ A_n:=\\sum_{k=0}^n\\binom nk^2\\binom{n+k}k^2. \\end{gather*} For $n=0,1,2,\\ldots$, we show that $6^{-n}|f_{i+j}|_{0\\leq i,j\\leq n}$ and $12^{-n}|D_{i+j}|_{0\\le i,j\\le n}$ are positive odd integers, and $10^{-n}|b_{i+j}|_{0\\leq i,j\\leq n}$ and $24^{-n}|A_{i+j}|_{0\\leq i,j\\leq n}$ are always integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-22T03:57:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A10",
      "11A07",
      "11B65",
      "11C20",
      "15B99"
    ],
    "keywords": [
      "combinatorial sequences",
      "hankel-type determinants",
      "apery numbers",
      "positive odd integers",
      "franel numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}