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 A281733 Positive integers T_i such that Sum_{k >= 0} (S_k * x^(2*k+1)) + 1/24 - Sum_{k >= 1} (T_k * x^(2*k)) = (cos((2/3) * arccos(6 * sqrt(3) * x)))/12 for all real x with |x| <= 1/(6 * sqrt(3)), where S_k = A176898(k). 2
 1, 32, 1792, 122880, 9371648, 763363328, 65028489216, 5722507051008, 516147694796800, 47463855386787840, 4433247375867248640, 419423751734223175680, 40109816011998942461952, 3870915577031009050296320, 376519953782381735485374464, 36874663860751966094632157184 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The terms are given on page 3 in Sun (2013). Conjecture: T_p == -2 (mod p) for any prime p (cf. Sun (2013), Conjecture 4). LINKS Davin Park, Table of n, a(n) for n = 1..100 K. H. Pilehrood and T. H. Pilehrood, Jacobi Polynomials and Congruences Involving Some Higher-Order Catalan Numbers and Binomial Coefficients, The Journal of Integer Sequences, 18 (2015), Article 15.11.7. Z. W. Sun, Products and sums divisible by central binomial coefficients, The Electronic Journal of Combinatorics, 20(1) (2013), #P9. FORMULA a(n) = 16^(n-1) * binomial(3*n-2, 2*n-1)/n. - Sarah Selkirk, Feb 11 2020 From Stefano Spezia, Feb 11 2020: (Start) O.g.f.: (1/24)*(1 - cos((2/3) * arcsin(6 * sqrt(3*x)))). E.g.f.: (1/24)*(1 - F([-1/3, 1/3], [1/2, 1], 108*x)), where F is the generalized hypergeometric function. (End) MATHEMATICA CoefficientList[Series[(1/24)(1 - Cos[(2/3) ArcSin[6 Sqrt[3x]]]), {x, 0, 20}], x] // Rest (* Davin Park, Feb 06 2017, updated by Jean-François Alcover, Mar 21 2020 *) CoefficientList[Series[(1-HypergeometricPFQ[{-1/3, 1/3}, {1/2, 1}, 108x])/24, {x, 0, 16}], x]*Table[n!, {n, 0, 16}] (* Stefano Spezia, Mar 21 2020 *) CROSSREFS Cf. A176898. Sequence in context: A183242 A186019 A186011 * A281835 A299079 A299842 Adjacent sequences: A281730 A281731 A281732 * A281734 A281735 A281736 KEYWORD nonn AUTHOR Felix Fröhlich, Jan 31 2017 EXTENSIONS Extended by Davin Park, Feb 06 2017 STATUS approved

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Last modified June 6 22:19 EDT 2023. Contains 363151 sequences. (Running on oeis4.)