A275454 G.f.: 3F2([1/9, 4/9, 8/9], [2/3, 1], 729 x).

1, 48, 15912, 7205484, 3731294385, 2082701917296, 1219626159039288, 738421413473848104, 458174434421099404008, 289681112497807349679360, 185894363292170517130962816, 120738965077159251405022531728, 79206198459248339865163888224660, 52397749335891513408552541281755520

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

Cf. A268545-A268555, A275051-A275054.

Sequence in context: A275570 A307618 A225786 * A202928 A265866 A159425

Adjacent sequences: A275451 A275452 A275453 * A275455 A275456 A275457

Keyword

nonn

Author

Gheorghe Coserea, Jul 31 2016

Status

approved