A275453 G.f.: 3F2([1/9, 4/9, 7/9], [2/3, 1], 729 x).

1, 42, 13104, 5705700, 2870226450, 1565667525240, 899552741658480, 535848881630582520, 327799728893143306800, 204660966917426732512800, 129859500691523648952466560, 83483493583251639541209993720, 54254332317972702411364923299700, 35581785531539194815959254026276000

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], [2/3, 1], 729*x).

a(n) = 729^n*Gamma(2/3)*Gamma(1/9+n)*Gamma(4/9+n)*Gamma(7/9+n)/((n!)^2*Gamma(1/9)*Gamma(4/9)*Gamma(7/9)*Gamma(2/3+n)). - Benedict W. J. Irwin, Aug 05 2016

D-finite with recurrence n^2*(3*n-1)*a(n) -3*(9*n-5)*(9*n-8)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Jul 27 2022

Example

1 + 42*x + 13104*x^2 + 5705700*x^3 + ...

Mathematica

a[n_] := FullSimplify[(729^n Gamma[2/3] Gamma[1/9 + n] Gamma[4/9 + n] Gamma[7/9 + n])/((n!)^2 Gamma[1/9] Gamma[4/9] Gamma[7/9] Gamma[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, 4/9, 7/9], [2/3, 1], 729*x, N))
(PARI) a(n) = round(729^n*gamma(2/3)*gamma(1/9+n)*gamma(4/9+n)*gamma(7/9+n)/((n!)^2*gamma(1/9)*gamma(4/9)*gamma(7/9)*gamma(2/3+n))) \\ Charles R Greathouse IV, Aug 05 2016

Crossrefs

Cf. A268545-A268555, A275051-A275054.

Sequence in context: A289396 A159417 A289540 * A268552 A227493 A095423

Adjacent sequences: A275450 A275451 A275452 * A275454 A275455 A275456

Keyword

nonn

Author

Gheorghe Coserea, Jul 30 2016

Status

approved