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%I #13 Jul 27 2022 06:54:37
%S 1,42,13104,5705700,2870226450,1565667525240,899552741658480,
%T 535848881630582520,327799728893143306800,204660966917426732512800,
%U 129859500691523648952466560,83483493583251639541209993720,54254332317972702411364923299700,35581785531539194815959254026276000
%N G.f.: 3F2([1/9, 4/9, 7/9], [2/3, 1], 729 x).
%C "Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).
%H Gheorghe Coserea, <a href="/A275453/b275453.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, 4/9, 7/9], [2/3, 1], 729*x).
%F a(n) = 729^n*Gamma(2/3)*Gamma(1/9+n)*Gamma(4/9+n)*Gamma(7/9+n)/((n!)^2*Gamma(1/9)*Gamma(4/9)*Gamma(7/9)*Gamma(2/3+n)). - _Benedict W. J. Irwin_, Aug 05 2016
%F D-finite with recurrence n^2*(3*n-1)*a(n) -3*(9*n-5)*(9*n-8)*(9*n-2)*a(n-1)=0. - _R. J. Mathar_, Jul 27 2022
%e 1 + 42*x + 13104*x^2 + 5705700*x^3 + ...
%t a[n_] := FullSimplify[(729^n Gamma[2/3] Gamma[1/9 + n] Gamma[4/9 + n] Gamma[7/9 + n])/((n!)^2 Gamma[1/9] Gamma[4/9] Gamma[7/9] 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, 4/9, 7/9], [2/3, 1], 729*x, N))
%o (PARI) a(n) = round(729^n*gamma(2/3)*gamma(1/9+n)*gamma(4/9+n)*gamma(7/9+n)/((n!)^2*gamma(1/9)*gamma(4/9)*gamma(7/9)*gamma(2/3+n))) \\ _Charles R Greathouse IV_, Aug 05 2016
%Y Cf. A268545-A268555, A275051-A275054.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Jul 30 2016
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