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A275452
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G.f.: 3F2([1/9, 4/9, 7/9], [1/3, 1], 729 x).
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1
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1, 84, 32760, 16302000, 9020711700, 5299182393120, 3234930051733380, 2028415806982164600, 1297264109283593451000, 842341453312777393815840, 553562736218491223861661024, 367351669654325623384052435136, 245756466255265144369306647476400
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OFFSET
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0,2
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COMMENTS
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"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).
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LINKS
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Gheorghe Coserea, Table of n, a(n) for n = 0..300
A. Bostan, S. Boukraa, G. Christol, S. Hassani, J-M. Maillard Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity, arXiv:1211.6031 [math-ph], 2012.
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FORMULA
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G.f.: hypergeom([1/9, 4/9, 7/9], [1/3, 1], 729*x).
From Vaclav Kotesovec, Jul 31 2016: (Start)
Recurrence: n^2*(3*n - 2)*a(n) = 3*(9*n - 8)*(9*n - 5)*(9*n - 2)*a(n-1).
a(n) ~ Gamma(1/3) * 3^(6*n) / (Gamma(1/9) * Gamma(4/9) * Gamma(7/9) * n).
a(n) ~ 2^(2/9) * Gamma(1/3) * sin(Pi/9) * 3^(6*n) / (sqrt(Pi) * Gamma(4/9) * Gamma(7/18) * n).
(End)
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
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EXAMPLE
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1 + 84*x + 32760*x^2 + ...
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MATHEMATICA
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CoefficientList[Series[HypergeometricPFQ[{1/9, 4/9, 7/9}, {1/3, 1}, 729*x], {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 31 2016 *)
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 *)
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PROG
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(PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
hypergeom([1/9, 4/9, 7/9], [1/3, 1], 729*x, N)
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CROSSREFS
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Cf. A268545-A268555, A275051-A275054.
Sequence in context: A202923 A232914 A145495 * A269933 A184126 A185403
Adjacent sequences: A275449 A275450 A275451 * A275453 A275454 A275455
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KEYWORD
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nonn
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AUTHOR
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Gheorghe Coserea, Jul 30 2016
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STATUS
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approved
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