{
  "id": "1407.3533",
  "version": "v4",
  "published": "2014-07-14T03:55:11.000Z",
  "updated": "2015-01-19T02:56:36.000Z",
  "title": "Generalising Tuenter's binomial sums",
  "authors": [
    "Richard P. Brent"
  ],
  "comment": "17 pages, 2 appendices, corrected typos in v2, added OEIS references in v3, corrected abstract in v4",
  "categories": [
    "math.CO"
  ],
  "abstract": "Tuenter [Fibonacci Quarterly 40 (2002), 175-180] and other authors have considered centred binomial sums of the form \\[S_r(n) = \\sum_k \\binom{2n}{k}|n-k|^r,\\] where $r$ and $n$ are non-negative integers. We consider sums of the form \\[U_r(n) = \\sum_k \\binom{n}{k}|n/2-k|^r\\] which are a generalisation of Tuenter's sums as $S_r(n) = U_r(2n)$ but $U_r(n)$ is also well-defined for odd arguments $n$. $U_r(n)$ may be interpreted as a moment of a symmetric Bernoulli random walk with $n$ steps. The form of $U_r(n)$ depends on the parities of both $r$ and $n$. In fact, $U_r(n)$ is the product of a polynomial (depending on the parities of $r$ and $n$) times a power of two or a binomial coefficient. In all cases the polynomials can be expressed in terms of Dumont-Foata polynomials. We give recurrence relations, generating functions and explicit formulas for the functions $U_r(n)$ and/or the associated polynomials.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-07-16T03:39:35.000Z",
      "abstract": "Tuenter [Fibonacci Quarterly 40 (2002), 175-180] and other authors have considered centred binomial sums of the form \\[S_r(n) = \\sum_k \\binom{2n}{k}|n-k|^r,\\] where $r$ and $n$ are non-negative integers. We consider sums of the form \\[U_r(n) = \\sum_k \\binom{n}{k}|n/2-k|^r\\] which are a generalisation of Tuenter's sums as $S_r(n) = U_r(2n)$ but $U_r(n)$ is also well-defined for odd arguments $n$. $U_r(n)$ may be interpreted as a moment of a symmetric Bernoulli random walk with $n$ steps. We give recurrence relations, generating functions and explicit formulas for the functions $U_r(n)$. The form of the solutions depends on the parities of both $r$ and $n$. When $r$ is even, $U_r(n)/4^n$ is a polynomial in $n$; when $r$ is odd, $U_r(n)/\\binom{2n}{n}$ is a polynomial in $n$. In all cases these polynomials can be expressed in terms of Dumont-Foata polynomials.",
      "comment": "17 pages, 2 appendices, corrected typos in v2, added OEIS references in v3",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-01-19T02:56:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A10",
      "11B65",
      "05A15",
      "05A19",
      "44A60",
      "60G50"
    ],
    "keywords": [
      "generalising tuenters binomial sums",
      "polynomial",
      "symmetric bernoulli random walk",
      "recurrence relations",
      "tuenters sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.3533B"
    }
  }
}