{
  "id": "1208.3903",
  "version": "v8",
  "published": "2012-08-19T23:30:05.000Z",
  "updated": "2014-12-23T04:53:07.000Z",
  "title": "On monotonicity of some combinatorial sequences",
  "authors": [
    "Qing-Hu Hou",
    "Zhi-Wei Sun",
    "Haomin Wen"
  ],
  "comment": "10 pages",
  "journal": "Publ. Math. Debrecen 85(2014), no.3-4, 285-295",
  "categories": [
    "math.CO"
  ],
  "abstract": "We confirm Sun's conjecture that $(\\root{n+1}\\of{F_{n+1}}/\\root{n}\\of{F_n})_{n\\ge 4}$ is strictly decreasing to the limit 1, where $(F_n)_{n\\ge0}$ is the Fibonacci sequence. We also prove that the sequence $(\\root{n+1}\\of{D_{n+1}}/\\root{n}\\of{D_n})_{n\\ge3}$ is strictly decreasing with limit $1$, where $D_n$ is the $n$-th derangement number. For $m$-th order harmonic numbers $H_n^{(m)}=\\sum_{k=1}^n 1/k^m\\ (n=1,2,3,\\ldots)$, we show that $(\\root{n+1}\\of{H^{(m)}_{n+1}}/\\root{n}\\of{H^{(m)}_n})_{n\\ge3}$ is strictly increasing.",
  "revisions": [
    {
      "version": "v7",
      "updated": "2014-04-10T14:24:02.000Z",
      "comment": "10 pages. Accepted version for publication in Publ. Math. Debrecen",
      "journal": null,
      "doi": null
    },
    {
      "version": "v8",
      "updated": "2014-12-23T04:53:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A10",
      "11B39",
      "11B75"
    ],
    "keywords": [
      "combinatorial sequences",
      "monotonicity",
      "th order harmonic numbers",
      "th derangement number",
      "confirm suns conjecture"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.3903H"
    }
  }
}