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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).

%I A281733 #54 Dec 09 2024 19:50:12
%S A281733 1,32,1792,122880,9371648,763363328,65028489216,5722507051008,
%T A281733 516147694796800,47463855386787840,4433247375867248640,
%U A281733 419423751734223175680,40109816011998942461952,3870915577031009050296320,376519953782381735485374464,36874663860751966094632157184
%N 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).
%C A281733 The terms are given on page 3 in Sun (2013).
%C A281733 Conjecture: T_p == -2 (mod p) for any prime p (cf. Sun (2013), Conjecture 4).
%C A281733 This is the odd bisection of A078531 divided by 2. The even bisection divided by 2 is A176898. - _Akiva Weinberger_, Dec 09 2024
%H A281733 Davin Park, <a href="/A281733/b281733.txt">Table of n, a(n) for n = 1..100</a>
%H A281733 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 A281733 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 A281733 a(n) = 16^(n-1) * binomial(3*n-2, 2*n-1)/n. - _Sarah Selkirk_, Feb 11 2020
%F A281733 From _Stefano Spezia_, Feb 11 2020: (Start)
%F A281733 O.g.f.: (1/24)*(1 - cos((2/3) * arcsin(6 * sqrt(3*x)))).
%F A281733 E.g.f.: (1/24)*(1 - F([-1/3, 1/3], [1/2, 1], 108*x)), where F is the generalized hypergeometric function. (End)
%F A281733 a(n) = binomial(6n-3, 3n-3/2)*binomial(3n-3/2, n-1/2)/(4*n*binomial(2*n-1, n-1/2)). - _Akiva Weinberger_, Dec 09 2024
%F A281733 a(n) = A078531(2*n-1)/2. - _Akiva Weinberger_, Dec 09 2024
%t A281733 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 A281733 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 A281733 Cf. A176898, A078531.
%K A281733 nonn
%O A281733 1,2
%A A281733 _Felix Fröhlich_, Jan 31 2017
%E A281733 Extended by _Davin Park_, Feb 06 2017