{
  "id": "math/0404385",
  "version": "v3",
  "published": "2004-04-21T14:24:55.000Z",
  "updated": "2008-07-14T15:58:25.000Z",
  "title": "On sums of binomial coefficients and their applications",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "journal": "Discrete Math. 308(2008), 4231-4245",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper we study recurrences concerning the combinatorial sum $[n,r]_m=\\sum_{k\\equiv r (mod m)}\\binom {n}{k}$ and the alternate sum $\\sum_{k\\equiv r (mod m)}(-1)^{(k-r)/m}\\binom{n}{k}$, where m>0, $n\\ge 0$ and r are integers. For example, we show that if $n\\ge m-1$ then $$\\sum_{i=0}^{\\lfloor(m-1)/2\\rfloor}(-1)^i\\binom{m-1-i}i [n-2i,r-i]_m=2^{n-m+1}.$$ We also apply such results to investigate Bernoulli and Euler polynomials. Our approach depends heavily on an identity established by the author [Integers 2(2002)].",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-07-14T15:58:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "05A19",
      "11B37",
      "11B68"
    ],
    "keywords": [
      "binomial coefficients",
      "applications",
      "euler polynomials",
      "alternate sum",
      "combinatorial sum"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......4385S"
    }
  }
}