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%I #15 Jul 27 2022 09:55:37
%S 1,21,5544,2194500,1032711750,535163031270,294927297193620,
%T 169625328357359160,100668944872954458000,61198401105544584882000,
%U 37917347562767179794006720,23857493242377754986443300520,15203471586481919338384641390900,9792887011999654848831359131972500
%N G.f.: 3F2([1/9, 2/9, 7/9], [2/3,1], 729x).
%C "Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).
%H Gheorghe Coserea, <a href="/A275053/b275053.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, 7/9], [2/3,1], 729*x).
%F From _Robert Israel_, Jul 28 2016: (Start)
%F a(n) = 729^n*Gamma(2/9+n)*sin((2/9)*Pi)*Gamma(1/9+n)*Gamma(7/9+n)*Gamma(2/3)/(Pi*Gamma(1/9)*Gamma(n+1)^2*Gamma(n+2/3)).
%F a(n+1) = a(n)*3*(2+9*n)*(1+9*n)*(7+9*n)/((n+1)^2*(3*n+2)).
%F (End)
%F a(n) ~ 2 * 3^(6*n - 1/2) * sin(2*Pi/9) / (Gamma(1/3) * Gamma(1/9) * n^(14/9)). - _Vaclav Kotesovec_, Jul 28 2016
%F D-finite with recurrence n^2*(3*n-1)*a(n) -3*(9*n-7)*(9*n-8)*(9*n-2)*a(n-1)=0. - _R. J. Mathar_, Jul 27 2022
%e 1 + 21*x + 5544*x^2 + 2194500*x^3 + ...
%p f:= gfun:-rectoproc({(-2187*n^3-8991*n^2-12042*n-5280)*a(n+1)+(3*n^3+17*n^2+32*n+20)*a(n+2), a(0) = 1, a(1) = 21}, a(n),remember):
%p map(f, [$0..20]); # _Robert Israel_, Jul 28 2016
%t CoefficientList[Series[HypergeometricPFQ[{1/9, 2/9, 7/9}, {2/3, 1}, 729 x], {x, 0, 13}], x] (* _Michael De Vlieger_, Jul 26 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, 7/9], [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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