A275452 G.f.: 3F2([1/9, 4/9, 7/9], [1/3, 1], 729 x).

1, 84, 32760, 16302000, 9020711700, 5299182393120, 3234930051733380, 2028415806982164600, 1297264109283593451000, 842341453312777393815840, 553562736218491223861661024, 367351669654325623384052435136, 245756466255265144369306647476400

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

From Vaclav Kotesovec, Jul 31 2016: (Start)

Recurrence: n^2*(3*n - 2)*a(n) = 3*(9*n - 8)*(9*n - 5)*(9*n - 2)*a(n-1).

a(n) ~ Gamma(1/3) * 3^(6*n) / (Gamma(1/9) * Gamma(4/9) * Gamma(7/9) * n).

a(n) ~ 2^(2/9) * Gamma(1/3) * sin(Pi/9) * 3^(6*n) / (sqrt(Pi) * Gamma(4/9) * Gamma(7/18) * n).

(End)

a(n) = (729^n * Gamma(1/3) * Gamma(1/9 + n) * Gamma(4/9+n) * Gamma(7/9 + n))/(n!^2*Gamma(1/9) * Gamma(4/9) * Gamma(7/9) * Gamma(1/3 + n)). - Benedict W. J. Irwin, Aug 09 2016

Example

1 + 84*x + 32760*x^2 + ...

Mathematica

CoefficientList[Series[HypergeometricPFQ[{1/9, 4/9, 7/9}, {1/3, 1}, 729*x], {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 31 2016 *)
FullSimplify[Table[(729^n Gamma[1/3] Gamma[1/9 + n] Gamma[4/9 + n] Gamma[7/9 + n])/((n!)^2 Gamma[1/9] Gamma[4/9] Gamma[7/9] Gamma[1/3 + n]), {n, 0, 20}]] (* Benedict W. J. Irwin, Aug 09 2016 *)

Prog

(PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
hypergeom([1/9, 4/9, 7/9], [1/3, 1], 729*x, N)

Crossrefs

Cf. A268545-A268555, A275051-A275054.

Sequence in context: A202923 A232914 A145495 * A269933 A184126 A185403

Adjacent sequences: A275449 A275450 A275451 * A275453 A275454 A275455

Keyword

nonn

Author

Gheorghe Coserea, Jul 30 2016

Status

approved