{
  "id": "1008.3887",
  "version": "v13",
  "published": "2010-08-23T19:07:36.000Z",
  "updated": "2014-06-17T14:52:55.000Z",
  "title": "Congruences involving generalized central trinomial coefficients",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "34 pages",
  "journal": "Sci. China Math. 57(2014), 1375-1400",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For integers $b$ and $c$ the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Those $T_n=T_n(1,1)\\ (n=0,1,2,\\ldots)$ are the usual central trinomial coefficients, and $T_n(3,2)$ coincides with the Delannoy number $D_n=\\sum_{k=0}^n\\binom nk\\binom{n+k}k$ in combinatorics. We investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each $n=1,2,3,\\ldots$ we have $$\\sum_{k=0}^{n-1}(2k+1)T_k(b,c)^2(b^2-4c)^{n-1-k}\\equiv0\\pmod{n^2}$$ and in particular $n^2\\mid\\sum_{k=0}^{n-1}(2k+1)D_k^2$; if $p$ is an odd prime then $$\\sum_{k=0}^{p-1}T_k^2\\equiv\\left(\\frac{-1}p\\right)\\ \\pmod{p}\\ \\ \\ {\\rm and}\\ \\ \\ \\sum_{k=0}^{p-1}D_k^2\\equiv\\left(\\frac 2p\\right)\\ \\pmod{p},$$ where $(-)$ denotes the Legendre symbol. We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.",
  "revisions": [
    {
      "version": "v13",
      "updated": "2014-06-17T14:52:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B75",
      "05A15",
      "11E25"
    ],
    "keywords": [
      "congruences",
      "usual central trinomial coefficients",
      "binary quadratic forms",
      "odd prime",
      "delannoy number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.3887S"
    }
  }
}