A275053 G.f.: 3F2([1/9, 2/9, 7/9], [2/3,1], 729x).

1, 21, 5544, 2194500, 1032711750, 535163031270, 294927297193620, 169625328357359160, 100668944872954458000, 61198401105544584882000, 37917347562767179794006720, 23857493242377754986443300520, 15203471586481919338384641390900, 9792887011999654848831359131972500

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, 7/9], [2/3,1], 729*x).

From Robert Israel, Jul 28 2016: (Start)

a(n) = 729^n*Gamma(2/9+n)*sin((2/9)*Pi)*Gamma(1/9+n)*Gamma(7/9+n)*Gamma(2/3)/(Pi*Gamma(1/9)*Gamma(n+1)^2*Gamma(n+2/3)).

a(n+1) = a(n)*3*(2+9*n)*(1+9*n)*(7+9*n)/((n+1)^2*(3*n+2)).

(End)

a(n) ~ 2 * 3^(6*n - 1/2) * sin(2*Pi/9) / (Gamma(1/3) * Gamma(1/9) * n^(14/9)). - Vaclav Kotesovec, Jul 28 2016

D-finite with recurrence n^2*(3*n-1)*a(n) -3*(9*n-7)*(9*n-8)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Jul 27 2022

Example

1 + 21*x + 5544*x^2 + 2194500*x^3 + ...

Maple

f:= gfun:-rectoproc({(-2187*n^3-8991*n^2-12042*n-5280)*a(n+1)+(3*n^3+17*n^2+32*n+20)*a(n+2), a(0) = 1, a(1) = 21}, a(n), remember):
map(f, [$0..20]); # Robert Israel, Jul 28 2016

Mathematica

CoefficientList[Series[HypergeometricPFQ[{1/9, 2/9, 7/9}, {2/3, 1}, 729 x], {x, 0, 13}], x] (* Michael De Vlieger, Jul 26 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, 7/9], [2/3, 1], 729*x, N))

Crossrefs

Cf. A268545-A268555.

Sequence in context: A232344 A074053 A220552 * A028451 A178327 A231906

Adjacent sequences: A275050 A275051 A275052 * A275054 A275055 A275056

Keyword

nonn

Author

Gheorghe Coserea, Jul 20 2016

Status

approved