A275456 G.f.: 3F2([1/9, 7/9, 8/9], [1/3, 1], 729 x).
1, 168, 85680, 50388000, 31479903000, 20342022734880, 13431668094985140, 9002968680250888200, 6101557410115488321000, 4170391891453158061891200, 2869634745103513910507157888, 1985363415926004500849300108544, 1379778913200535726019164327886400, 962553011288199733460143650698784000
Offset: 0
Keywords
Examples
1 + 168*x + 85680*x^2 + 50388000*x^3 + ...
Links
- 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.
Programs
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Mathematica
FullSimplify[Table[(729^n Gamma[1/3]Gamma[1/9+n]Gamma[7/9+n]Gamma[8/9+n]Sin[Pi/9]) / (Pi n!^2Gamma[7/9]Gamma[1/3+n]), {n, 0, 20}]] (* Benedict W. J. Irwin, Aug 10 2016 *) CoefficientList[Series[HypergeometricPFQ[{1/9, 7/9, 8/9}, {1/3, 1}, 729*x], {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 13 2016 *) -
PARI
\\ system("wget http://www.jjj.de/pari/hypergeom.gpi"); read("hypergeom.gpi"); N = 12; x = 'x + O('x^N); Vec(hypergeom([1/9, 7/9, 8/9], [1/3, 1], 729*x, N))
Formula
G.f.: hypergeom([1/9, 7/9, 8/9], [1/3, 1], 729*x).
a(n) = (729^n*Gamma(1/3)*Gamma(1/9+n)*Gamma(7/9+n)*Gamma(8/9+n)*sin(Pi/9)) / (Pi*n!^2*Gamma(7/9)*Gamma(1/3+n)). - Benedict W. J. Irwin, Aug 10 2016
a(n) ~ 2*sin(Pi/9)*3^(6*n-1/2) / (Gamma(2/3)*Gamma(7/9)*n^(5/9)). - Vaclav Kotesovec, Aug 13 2016
D-finite with recurrence n^2*(3*n-2)*a(n) -3*(9*n-8)*(9*n-2)*(9*n-1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022
Comments