A275457 - OEIS

%I #22 Nov 19 2024 02:41:17

%S 1,120,45045,21707400,11708971560,6735720993408,4039678502036100,

%T 2494516661768577600,1573990406710539567750,1009797626141015909237040,

%U 656436978973434195655059942,431326871057383042747830748560,285942228994752084893009228453460,190985447073724962020463006948873600

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

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

%H Gheorghe Coserea, <a href="/A275457/b275457.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([2/9, 4/9, 5/9], [1/3, 1], 729*x).

%F From _Robert Israel_, Jan 20 2017: (Start)

%F 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)).

%F 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)).

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

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

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

%p A[0]:= 1:

%p 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:

%p seq(A[i],i=0..21); # _Robert Israel_, Jan 20 2017

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

%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([2/9, 4/9, 5/9], [1/3, 1], 729*x, N))

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

%K nonn,changed

%O 0,2

%A _Gheorghe Coserea_, Jul 31 2016