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%I #50 Aug 08 2020 01:42:32
%S 1,32,1792,122880,9371648,763363328,65028489216,5722507051008,
%T 516147694796800,47463855386787840,4433247375867248640,
%U 419423751734223175680,40109816011998942461952,3870915577031009050296320,376519953782381735485374464,36874663860751966094632157184
%N 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).
%C The terms are given on page 3 in Sun (2013).
%C Conjecture: T_p == -2 (mod p) for any prime p (cf. Sun (2013), Conjecture 4).
%H Davin Park, <a href="/A281733/b281733.txt">Table of n, a(n) for n = 1..100</a>
%H K. H. Pilehrood and T. H. Pilehrood, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Pilehrood/pile5.html">Jacobi Polynomials and Congruences Involving Some Higher-Order Catalan Numbers and Binomial Coefficients</a>, The Journal of Integer Sequences, 18 (2015), Article 15.11.7.
%H Z. W. Sun, <a href="https://doi.org/10.37236/3022">Products and sums divisible by central binomial coefficients</a>, The Electronic Journal of Combinatorics, 20(1) (2013), #P9.
%F a(n) = 16^(n-1) * binomial(3*n-2, 2*n-1)/n. - _Sarah Selkirk_, Feb 11 2020
%F From _Stefano Spezia_, Feb 11 2020: (Start)
%F O.g.f.: (1/24)*(1 - cos((2/3) * arcsin(6 * sqrt(3*x)))).
%F E.g.f.: (1/24)*(1 - F([-1/3, 1/3], [1/2, 1], 108*x)), where F is the generalized hypergeometric function. (End)
%t 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 *)
%t 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 *)
%Y Cf. A176898.
%K nonn
%O 1,2
%A _Felix Fröhlich_, Jan 31 2017
%E Extended by _Davin Park_, Feb 06 2017
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