|
%I #24 Oct 08 2016 06:11:31
%S 1,5,43,461,5491,69395,910855,12274925,168668035,2352544535,
%T 33204000853,473179375355,6797163712639,98299113206663,
%U 1429765398030943,20899401842991341,306819063154144675,4521526749077118143,66858281393757281641,991598171159871109391
%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+5)*(2*x-1)*(x^2-47*x+3)*Dx^2 + (12*x^4-340*x^3-1319*x^2+530*x-15)*Dx + 4*x^3-24*x^2-445*x+75.
%H Gheorghe Coserea, <a href="/A274666/b274666.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^5*(x^2-47*x+3)*(-1+2*x)^2/(1-20*x+78*x^2-44*x^3+x^4)^3)/(1-20*x+78*x^2-44*x^3+x^4)^(1/4).
%F 0 = x*(2*x+5)*(2*x-1)*(x^2-47*x+3)*y'' + (12*x^4-340*x^3-1319*x^2+530*x-15)*y' + (4*x^3-24*x^2-445*x+75)*y, where y is the g.f.
%F Recurrence: 3*n^2*(39*n - 64)*a(n) = (2067*n^3 - 5459*n^2 + 3947*n - 930)*a(n-1) - (3705*n^3 - 13490*n^2 + 15323*n - 5230)*a(n-2) + 2*(n-2)^2*(39*n - 25)*a(n-3). - _Vaclav Kotesovec_, Jul 05 2016
%F a(n) ~ sqrt(53 + 191/sqrt(13)) * (47 + 13*sqrt(13))^n / (sqrt(2)*Pi*n*6^(n+1)). - _Vaclav Kotesovec_, Jul 05 2016
%t CoefficientList[Series[HypergeometricPFQ[{1/12, 5/12},{1},1728*x^5*(x^2-47*x+3)*(-1+2*x)^2/(1-20*x+78*x^2-44*x^3+x^4)^3]/(1-20*x+78*x^2-44*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 = 20; x = 'x + O('x^N);
%o Vec(hypergeom([1/12, 5/12],[1],1728*x^5*(x^2-47*x+3)*(-1+2*x)^2/(1-20*x+78*x^2-44*x^3+x^4)^3, N)/(1-20*x+78*x^2-44*x^3+x^4)^(1/4))
%Y Cf. A268545-A268555.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Jul 02 2016
| 