A275457 G.f.: 3F2([2/9, 4/9, 5/9], [1/3, 1], 729 x).

1, 120, 45045, 21707400, 11708971560, 6735720993408, 4039678502036100, 2494516661768577600, 1573990406710539567750, 1009797626141015909237040, 656436978973434195655059942, 431326871057383042747830748560, 285942228994752084893009228453460, 190985447073724962020463006948873600

Offset: 0

Author

Gheorghe Coserea, Jul 31 2016

Keywords

• nonn

Comments

"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

Examples

			1 + 120*x + 45045*x^2 + 21707400*x^3 + ...
		

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.

Crossrefs

Cf. A007949, A053735, A268545-A268555, A275051-A275054.

Programs

Maple

A[0]:= 1:
for n from 0 to 20 do A[n+1]:= 3*(5+9*n)*(2+9*n)*(4+9*n)*A[n]/((n+1)^2*(3*n+1)) od:
seq(A[i],i=0..21); # Robert Israel, Jan 20 2017

Mathematica

CoefficientList[HypergeometricPFQ[{2/9, 4/9, 5/9}, {1/3, 1}, 729 x] + O[x]^14, x] (* Jean-François Alcover, Sep 18 2018 *)

PARI

\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([2/9, 4/9, 5/9], [1/3, 1], 729*x, N))

Formula

G.f.: hypergeom([2/9, 4/9, 5/9], [1/3, 1], 729*x).

From Robert Israel, Jan 20 2017: (Start)

a(n) = (2/3)*729^n*Gamma(5/9+n)*Gamma(2/9+n)*Gamma(4/9+n)*sin((4/9)*Pi)*3^(1/2)/(Gamma(2/9)*Gamma(n+1)^2*Gamma(n+1/3)*Gamma(2/3)).

D-finite with recurrence a(n+1) = 3*(5+9*n)*(2+9*n)*(4+9*n)*a(n)/((n+1)^2*(3*n+1)).

a(n) ~ (2*sin(4*Pi/9)/(sqrt(3)*Gamma(2/9)*Gamma(2/3)))*729^n/n^(10/9).

A007949(a(n)) = A053735(n). (End)