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%I #13 Aug 07 2016 22:29:28
%S 1,24,6732,2771340,1342525275,711891288108,399866544799722,
%T 233750557331494632,140707672445849703480,86621407014527646518400,
%U 54278825541246092520182592,34504174655166790354911360048,22195631874904018057471849288020,14421008706115620277976088538033200
%N G.f.: 3F2([1/9, 2/9, 8/9], [2/3,1], 729 x).
%C "Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).
%H Gheorghe Coserea, <a href="/A275054/b275054.txt">Table of n, a(n) for n = 0..200</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, 2/9, 8/9], [2/3,1], 729*x).
%F From _Vaclav Kotesovec_, Jul 28 2016: (Start)
%F Recurrence: n^2*(3*n - 1)*a(n) = 3*(9*n - 8)*(9*n - 7)*(9*n - 1)*a(n-1).
%F a(n) ~ 2 * sin(Pi/9) * 3^(6*n - 1/2) / (Gamma(1/3) * Gamma(2/9) * n^(13/9)).
%F (End)
%F a(n) = 729^n*Gamma(2/3)*Gamma(1/9+n)*Gamma(2/9+n)*Gamma(8/9+n)*Sin(Pi/9)/(Pi*(n!)^2*Gamma(2/9)*Gamma(2/3+n)). - _Benedict W. J. Irwin_, Aug 05 2016
%e 1 + 24*x + 6732*x^2 + 2771340*x^3 + ...
%t CoefficientList[Series[HypergeometricPFQ[{1/9, 2/9, 8/9}, {2/3, 1}, 729 x], {x, 0, 13}], x] (* _Michael De Vlieger_, Jul 26 2016 *)
%t a[n_] := FullSimplify[(729^n Gamma[2/3] Gamma[1/9 + n] Gamma[2/9 + n] Gamma[8/9 + n] Sin[Pi/9])/(Pi (n!)^2 Gamma[2/9] Gamma[2/3 + n])] (* _Benedict W. J. Irwin_, Aug 05 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, 2/9, 8/9], [2/3,1], 729*x, N))
%Y Cf. A268545-A268555.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Jul 20 2016
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