A275454 - OEIS

%I #20 Jul 27 2022 05:57:23

%S 1,48,15912,7205484,3731294385,2082701917296,1219626159039288,

%T 738421413473848104,458174434421099404008,289681112497807349679360,

%U 185894363292170517130962816,120738965077159251405022531728,79206198459248339865163888224660,52397749335891513408552541281755520

%N G.f.: 3F2([1/9, 4/9, 8/9], [2/3, 1], 729 x).

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

%H Gheorghe Coserea, <a href="/A275454/b275454.txt">Table of n, a(n) for n = 0..300</a>

%H A. Bostan, S. Boukraa, G. Christol, S. Hassani, J-M. Maillard <a href="http://arxiv.org/abs/1211.6031">Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity</a>, arXiv:1211.6031 [math-ph], 2012.

%F G.f.: hypergeom([1/9, 4/9, 8/9], [2/3, 1], 729*x).

%F a(n) = (729^n*Gamma(2/3)*Gamma(1/9+n)*Gamma(4/9+n)*Gamma(8/9+n)*sin(Pi/9)) / (Pi*n!^2*Gamma(4/9)*Gamma(2/3+n)). - _Benedict W. J. Irwin_, Aug 09 2016

%F a(n) ~ 2*sin(Pi/9)*3^(6*n-1/2) / (Gamma(1/3)*Gamma(4/9)*n^(11/9)). - _Vaclav Kotesovec_, Aug 10 2016

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

%e 1 + 48*x + 15912*x^2 + 7205484*x^3 + ...

%t FullSimplify[Table[(729^n Gamma[2/3] Gamma[1/9 + n] Gamma[4/9 + n] Gamma[8/9 + n] Sin[Pi/9])/(Pi (n!)^2 Gamma[4/9] Gamma[2/3 + n]), {n, 0, 20}]] (* _Benedict W. J. Irwin_, Aug 09 2016 *)

%t CoefficientList[Series[HypergeometricPFQ[{1/9, 4/9, 8/9}, {2/3, 1}, 729*x], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Aug 10 2016 *)

%o (PARI)  \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");

%o read("hypergeom.gpi");

%o N = 12; x = 'x + O('x^N);

%o Vec(hypergeom([1/9, 4/9, 8/9], [2/3, 1], 729*x, N))

%Y Cf. A268545-A268555, A275051-A275054.

%K nonn

%O 0,2

%A _Gheorghe Coserea_, Jul 31 2016