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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
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
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
KEYWORD
nonn
AUTHOR
Felix Fröhlich, Jan 31 2017
EXTENSIONS
Extended by Davin Park, Feb 06 2017
STATUS
approved