A274667 - OEIS

%I #15 Oct 08 2016 06:11:24

%S 1,3,31,339,4131,53013,705139,9618003,133672387,1884947073,

%T 26889061761,387207732453,5619687743151,82101265925409,

%U 1206262382507451,17809706204128659,264074421220475427,3930338612143125849,58692717332813782501,879093138034007102289,13202346737893575996541

%N Diagonal of the rational function 1/(1 - x - y - x y - x z - y z + x y z).

%C Annihilating differential operator: x*(2*x+1)*(6*x^2+x-8)*(x^3-41*x^2-29*x+2)*Dx^2 + (36*x^6-964*x^5-917*x^4+2394*x^3+2339*x^2+400*x-16)*Dx + 12*x^5-104*x^4+57*x^3+1067*x^2+640*x+48.

%H Gheorghe Coserea, <a href="/A274667/b274667.txt">Table of n, a(n) for n = 0..310</a>

%H A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015.

%H Jacques-Arthur Weil, <a href="http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/">Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"</a>

%F G.f.: hypergeom([1/12, 5/12],[1],1728*x^4*(x^3-41*x^2-29*x+2)*(1+2*x)^2/(1-12*x-34*x^2-36*x^3+x^4)^3)/(1-12*x-34*x^2-36*x^3+x^4)^(1/4).

%F 0 = x*(2*x+1)*(6*x^2+x-8)*(x^3-41*x^2-29*x+2)*y'' + (36*x^6-964*x^5-917*x^4+2394*x^3+2339*x^2+400*x-16)*y' + (12*x^5-104*x^4+57*x^3+1067*x^2+640*x+48)*y, where y(x) is the g.f.

%F Recurrence: 2*n^2*(469*n^2 - 2106*n + 2229)*a(n) = (11725*n^4 - 64375*n^3 + 111011*n^2 - 68153*n + 13344)*a(n-1) + (46431*n^4 - 301356*n^3 + 678782*n^2 - 620403*n + 186048)*a(n-2) + (37989*n^4 - 284553*n^3 + 757682*n^2 - 829732*n + 299712)*a(n-3) - 2*(n-3)^2*(469*n^2 - 1168*n + 592)*a(n-4). - _Vaclav Kotesovec_, Jul 05 2016

%t CoefficientList[Series[HypergeometricPFQ[{1/12, 5/12},{1},1728*x^4*(x^3-41*x^2-29*x+2)*(1+2*x)^2/(1-12*x-34*x^2-36*x^3+x^4)^3]/(1-12*x-34*x^2-36*x^3+x^4)^(1/4), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jul 05 2016 *)

%o (PARI)

%o my(x='x, y='y, z='z);

%o R = 1/(1 - x - y - x*y - x*z - y*z + x*y*z);

%o diag(n, expr, var) = {

%o   my(a = vector(n));

%o   for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));

%o   for (k = 1, n, a[k] = expr;

%o        for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));

%o   return(a);

%o };

%o diag(10, R, [x, y, z])

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

%o read("hypergeom.gpi");

%o N = 22; x = 'x + O('x^N);

%o Vec(hypergeom([1/12, 5/12],[1],1728*x^4*(x^3-41*x^2-29*x+2)*(1+2*x)^2/(1-12*x-34*x^2-36*x^3+x^4)^3, N)/(1-12*x-34*x^2-36*x^3+x^4)^(1/4))

%Y Cf. A268545-A268555.

%K nonn

%O 0,2

%A _Gheorghe Coserea_, Jul 02 2016