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

0,2

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, 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

Example

1 + 168*x + 85680*x^2 + 50388000*x^3 + ...

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 *)

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, 7/9, 8/9], [1/3, 1], 729*x, N))

Crossrefs

Cf. A268545-A268555, A275051-A275054.

Sequence in context: A130215 A275460 A364178 * A185404 A146200 A159394

Adjacent sequences: A275453 A275454 A275455 * A275457 A275458 A275459

Keyword

nonn

Author

Gheorghe Coserea, Jul 31 2016

Status

approved