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G.f.: 3F2([1/7, 2/7, 4/7], [1/2, 1], 2401 x).
1

%I #16 Aug 07 2016 22:29:21

%S 1,112,103488,139087872,219932697600,380982080962560,

%T 699690909055057920,1338362619711643975680,2637829075787918298316800,

%U 5319794376634771700187136000,10925401705883689450455905075200,22771065347191895949498972005990400,48042740185717267168321727861725593600

%N G.f.: 3F2([1/7, 2/7, 4/7], [1/2, 1], 2401 x).

%C "Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

%H Gheorghe Coserea, <a href="/A275052/b275052.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([1/7, 2/7, 4/7], [1/2, 1], 2401*x).

%F From _Vaclav Kotesovec_, Jul 28 2016: (Start)

%F Recurrence: n^2*(2*n - 1)*a(n) = 14*(7*n - 6)*(7*n - 5)*(7*n - 3)*a(n-1).

%F a(n) ~ sqrt(Pi) * 7^(4*n) / (Gamma(1/7) * Gamma(2/7) * Gamma(4/7) * n^(3/2)).

%F (End)

%F a(n) = 9604^n*Gamma(1/7+n)*Gamma(2/7+n)*Gamma(4/7+n)/(n!*(2n)!*Gamma(1/7)*Gamma(2/7)*Gamma(4/7)). - _Benedict W. J. Irwin_, Aug 05 2016

%e 1 + 112*x + 103488*x^2 + 139087872*x^3 + ...

%t CoefficientList[Series[HypergeometricPFQ[{1/7, 2/7, 4/7}, {1/2, 1}, 2401 x], {x, 0, 12}], x] (* _Michael De Vlieger_, Jul 26 2016 *)

%t a[n_] := FullSimplify[(9604^n Gamma[1/7 + n] Gamma[2/7 + n] Gamma[4/7 + n])/(n!(2n)! Gamma[1/7] Gamma[2/7] Gamma[4/7])] (* _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([1/7, 2/7, 4/7], [1/2, 1], 2401*x, N))

%Y Cf. A268545-A268555.

%K nonn

%O 0,2

%A _Gheorghe Coserea_, Jul 20 2016