A275054 G.f.: 3F2([1/9, 2/9, 8/9], [2/3,1], 729 x).

1, 24, 6732, 2771340, 1342525275, 711891288108, 399866544799722, 233750557331494632, 140707672445849703480, 86621407014527646518400, 54278825541246092520182592, 34504174655166790354911360048, 22195631874904018057471849288020, 14421008706115620277976088538033200

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

Cf. A268545-A268555.

Sequence in context: A181749 A266312 A181232 * A088616 A158042 A297049

Adjacent sequences: A275051 A275052 A275053 * A275055 A275056 A275057

Keyword

nonn

Author

Gheorghe Coserea, Jul 20 2016

Status

approved