A275054 G.f.: 3F2([1/9, 2/9, 8/9], [2/3,1], 729 x).
1, 24, 6732, 2771340, 1342525275, 711891288108, 399866544799722, 233750557331494632, 140707672445849703480, 86621407014527646518400, 54278825541246092520182592, 34504174655166790354911360048, 22195631874904018057471849288020, 14421008706115620277976088538033200
Offset: 0
Keywords
Examples
1 + 24*x + 6732*x^2 + 2771340*x^3 + ...
Links
- Gheorghe Coserea, Table of n, a(n) for n = 0..200
- 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
-
Mathematica
CoefficientList[Series[HypergeometricPFQ[{1/9, 2/9, 8/9}, {2/3, 1}, 729 x], {x, 0, 13}], x] (* Michael De Vlieger, Jul 26 2016 *) a[n_] := FullSimplify[(729^n Gamma[2/3] Gamma[1/9 + n] Gamma[2/9 + n] Gamma[8/9 + n] Sin[Pi/9])/(Pi (n!)^2 Gamma[2/9] Gamma[2/3 + n])] (* Benedict W. J. Irwin, Aug 05 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, 2/9, 8/9], [2/3,1], 729*x, N))
Formula
G.f.: hypergeom([1/9, 2/9, 8/9], [2/3,1], 729*x).
From Vaclav Kotesovec, Jul 28 2016: (Start)
Recurrence: n^2*(3*n - 1)*a(n) = 3*(9*n - 8)*(9*n - 7)*(9*n - 1)*a(n-1).
a(n) ~ 2 * sin(Pi/9) * 3^(6*n - 1/2) / (Gamma(1/3) * Gamma(2/9) * n^(13/9)).
(End)
a(n) = 729^n*Gamma(2/3)*Gamma(1/9+n)*Gamma(2/9+n)*Gamma(8/9+n)*Sin(Pi/9)/(Pi*(n!)^2*Gamma(2/9)*Gamma(2/3+n)). - Benedict W. J. Irwin, Aug 05 2016
Comments