A275455 G.f.: 3F2([1/9, 5/9, 8/9], [1/3, 1], 729 x).
1, 120, 53550, 28973100, 17036182800, 10496595041856, 6664244456261700, 4320449008019199000, 2844426519643185378000, 1894935877560218667820800, 1274265873172890987907535424, 863426385292565961502380501120, 588738285265666300220495724048000, 403569219885941102398195162309056000
Offset: 0
Keywords
Examples
1 + 120*x + 53550*x^2 + 28973100*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
-
Mathematica
FullSimplify[Table[(729^n Gamma[1/3]Gamma[1/9+n]Gamma[5/9+n]Gamma[8/9+n]Sin[Pi/9])/(Pi n!^2Gamma[5/9]Gamma[1/3+n]), {n, 0, 20}]] (* Benedict W. J. Irwin, Aug 10 2016 *) CoefficientList[Series[HypergeometricPFQ[{1/9, 5/9, 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, 5/9, 8/9], [1/3, 1], 729*x, N))
Formula
G.f.: hypergeom([1/9, 5/9, 8/9], [1/3, 1], 729*x).
a(n) = (729^n*Gamma(1/3)*Gamma(1/9+n)*Gamma(5/9+n)*Gamma(8/9+n)*sin(Pi/9)) / (Pi*n!^2*Gamma(5/9)*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(5/9)*n^(7/9)). - Vaclav Kotesovec, Aug 13 2016
D-finite with recurrence n^2*(3*n-2)*a(n) -3*(9*n-4)*(9*n-8)*(9*n-1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022
Comments