OFFSET
0,2
COMMENTS
"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).
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.
FORMULA
G.f.: hypergeom([1/9, 4/9, 8/9], [2/3, 1], 729*x).
a(n) = (729^n*Gamma(2/3)*Gamma(1/9+n)*Gamma(4/9+n)*Gamma(8/9+n)*sin(Pi/9)) / (Pi*n!^2*Gamma(4/9)*Gamma(2/3+n)). - Benedict W. J. Irwin, Aug 09 2016
a(n) ~ 2*sin(Pi/9)*3^(6*n-1/2) / (Gamma(1/3)*Gamma(4/9)*n^(11/9)). - Vaclav Kotesovec, Aug 10 2016
D-finite with recurrence n^2*(3*n-1)*a(n) -3*(9*n-5)*(9*n-8)*(9*n-1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022
EXAMPLE
1 + 48*x + 15912*x^2 + 7205484*x^3 + ...
MATHEMATICA
FullSimplify[Table[(729^n Gamma[2/3] Gamma[1/9 + n] Gamma[4/9 + n] Gamma[8/9 + n] Sin[Pi/9])/(Pi (n!)^2 Gamma[4/9] Gamma[2/3 + n]), {n, 0, 20}]] (* Benedict W. J. Irwin, Aug 09 2016 *)
CoefficientList[Series[HypergeometricPFQ[{1/9, 4/9, 8/9}, {2/3, 1}, 729*x], {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 10 2016 *)
PROG
(PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([1/9, 4/9, 8/9], [2/3, 1], 729*x, N))
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Jul 31 2016
STATUS
approved