{
  "id": "1710.10884",
  "version": "v1",
  "published": "2017-10-30T11:49:07.000Z",
  "updated": "2017-10-30T11:49:07.000Z",
  "title": "Divisibility of binomial coefficients by powers of two",
  "authors": [
    "Lukas Spiegelhofer",
    "Michael Wallner"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For nonnegative integers $j$ and $n$ let $\\Theta(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are not divisible by $2^{j+1}$. In this paper we prove that the family $j\\mapsto\\Theta(j,n)$ usually follows a normal distribution. The method used for proving this theorem involves the computation of first and second moments of $\\Theta(j,n)$, and uses asymptotic analysis of multivariate generating functions by complex analytic methods, building on earlier work by Drmota (1994) and Drmota, Kauers and Spiegelhofer (2016).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-30T11:49:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "05A16",
      "11A63",
      "11B50"
    ],
    "keywords": [
      "binomial coefficients",
      "divisibility",
      "complex analytic methods",
      "multivariate generating functions",
      "th row"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}