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%I #17 Aug 10 2016 03:42:05
%S 1,84,32760,16302000,9020711700,5299182393120,3234930051733380,
%T 2028415806982164600,1297264109283593451000,842341453312777393815840,
%U 553562736218491223861661024,367351669654325623384052435136,245756466255265144369306647476400
%N G.f.: 3F2([1/9, 4/9, 7/9], [1/3, 1], 729 x).
%C "Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).
%H Gheorghe Coserea, <a href="/A275452/b275452.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], [1/3, 1], 729*x).
%F From _Vaclav Kotesovec_, Jul 31 2016: (Start)
%F Recurrence: n^2*(3*n - 2)*a(n) = 3*(9*n - 8)*(9*n - 5)*(9*n - 2)*a(n-1).
%F a(n) ~ Gamma(1/3) * 3^(6*n) / (Gamma(1/9) * Gamma(4/9) * Gamma(7/9) * n).
%F a(n) ~ 2^(2/9) * Gamma(1/3) * sin(Pi/9) * 3^(6*n) / (sqrt(Pi) * Gamma(4/9) * Gamma(7/18) * n).
%F (End)
%F a(n) = (729^n * Gamma(1/3) * Gamma(1/9 + n) * Gamma(4/9+n) * Gamma(7/9 + n))/(n!^2*Gamma(1/9) * Gamma(4/9) * Gamma(7/9) * Gamma(1/3 + n)). - _Benedict W. J. Irwin_, Aug 09 2016
%e 1 + 84*x + 32760*x^2 + ...
%t CoefficientList[Series[HypergeometricPFQ[{1/9, 4/9, 7/9}, {1/3, 1}, 729*x], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jul 31 2016 *)
%t FullSimplify[Table[(729^n Gamma[1/3] Gamma[1/9 + n] Gamma[4/9 + n] Gamma[7/9 + n])/((n!)^2 Gamma[1/9] Gamma[4/9] Gamma[7/9] Gamma[1/3 + n]), {n, 0, 20}]] (* _Benedict W. J. Irwin_, Aug 09 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 hypergeom([1/9, 4/9, 7/9], [1/3, 1], 729*x, N)
%Y Cf. A268545-A268555, A275051-A275054.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Jul 30 2016
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