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%I #16 Jul 27 2022 06:00:26
%S 1,168,85680,50388000,31479903000,20342022734880,13431668094985140,
%T 9002968680250888200,6101557410115488321000,4170391891453158061891200,
%U 2869634745103513910507157888,1985363415926004500849300108544,1379778913200535726019164327886400,962553011288199733460143650698784000
%N G.f.: 3F2([1/9, 7/9, 8/9], [1/3, 1], 729 x).
%C "Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).
%H Gheorghe Coserea, <a href="/A275456/b275456.txt">Table of n, a(n) for n = 0..300</a>
%H A. Bostan, S. Boukraa, G. Christol, S. Hassani, J-M. Maillard <a href="http://arxiv.org/abs/1211.6031">Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity</a>, arXiv:1211.6031 [math-ph], 2012.
%F G.f.: hypergeom([1/9, 7/9, 8/9], [1/3, 1], 729*x).
%F a(n) = (729^n*Gamma(1/3)*Gamma(1/9+n)*Gamma(7/9+n)*Gamma(8/9+n)*sin(Pi/9)) / (Pi*n!^2*Gamma(7/9)*Gamma(1/3+n)). - _Benedict W. J. Irwin_, Aug 10 2016
%F a(n) ~ 2*sin(Pi/9)*3^(6*n-1/2) / (Gamma(2/3)*Gamma(7/9)*n^(5/9)). - _Vaclav Kotesovec_, Aug 13 2016
%F D-finite with recurrence n^2*(3*n-2)*a(n) -3*(9*n-8)*(9*n-2)*(9*n-1)*a(n-1)=0. - _R. J. Mathar_, Jul 27 2022
%e 1 + 168*x + 85680*x^2 + 50388000*x^3 + ...
%t FullSimplify[Table[(729^n Gamma[1/3]Gamma[1/9+n]Gamma[7/9+n]Gamma[8/9+n]Sin[Pi/9]) / (Pi n!^2Gamma[7/9]Gamma[1/3+n]), {n, 0, 20}]] (* _Benedict W. J. Irwin_, Aug 10 2016 *)
%t CoefficientList[Series[HypergeometricPFQ[{1/9, 7/9, 8/9}, {1/3, 1}, 729*x], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Aug 13 2016 *)
%o (PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
%o read("hypergeom.gpi");
%o N = 12; x = 'x + O('x^N);
%o Vec(hypergeom([1/9, 7/9, 8/9], [1/3, 1], 729*x, N))
%Y Cf. A268545-A268555, A275051-A275054.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Jul 31 2016
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