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..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.
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
EXAMPLE
1 + 24*x + 6732*x^2 + 2771340*x^3 + ...
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 *)
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, 2/9, 8/9], [2/3, 1], 729*x, N))
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Jul 20 2016
STATUS
approved