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 A178808 a(n) = (Sum_{k=0..n-1} (2*k+1)*(D_k)^2)/n^2, where D_0, D_1, ... are central Delannoy numbers given by A001850. 6
 1, 7, 97, 1791, 38241, 892039, 22092673, 571387903, 15271248769, 418796912007, 11725812711009, 333962374092543, 9648543623050593, 282164539499639559, 8338391167566634497, 248661515283002490879, 7474768663941435203073 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS On Jun 14 2010, Zhi-Wei Sun conjectured that a(n)=(Sum_{k=0}^{n-1} (2*k+1)*(D_k)^2)/n^2 is always an integer and that a(p) = p^2 - 4*p^3*q_p(2) - 2*p^4*q_p(2)^2 (mod p^5) for any prime p>3, where q_p(2) denotes the Fermat quotient (2^{p-1}-1)/p. He also conjectured that Sum_{k=0..n-1}(2*k+1)*(-1)^k*(D_k)^2 = 0 (mod n*D_n/(3,D_n)) for all n=1,2,3,.... LINKS G. C. Greubel, Table of n, a(n) for n = 1..500 Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers arXiv:1006.2776 [math.NT], 2011. FORMULA a(n) ~ (1 + sqrt(2))^(4*n) / (16*Pi*n^2). - Vaclav Kotesovec, Jan 24 2019 EXAMPLE For n=3 we have a(3) = (D_0^2 + 3D_1^2 + 5D_2^2)/3^2 = (1 + 3*3^2 + 5*13^2)/3^2 = 97. MATHEMATICA DD[n_]:=Sum[Binomial[n+k, 2k]Binomial[2k, k], {k, 0, n}]; SS[n_]:= Sum[(2k+1)*DD[k]^2, {k, 0, n-1}]/n^2; Table[SS[n], {n, 1, 25}] Table[Sum[(2k+1)*JacobiP[k, 0, 0, 3]^2, {k, 0, n-1}]/n^2, {n, 1, 30}] (* G. C. Greubel, Jan 23 2019 *) CROSSREFS Cf. A001850, A178790, A178791, A173774. Sequence in context: A011943 A218669 A188441 * A083083 A022007 A174516 Adjacent sequences:  A178805 A178806 A178807 * A178809 A178810 A178811 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jun 16 2010 STATUS approved

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Last modified June 1 00:27 EDT 2020. Contains 334756 sequences. (Running on oeis4.)