%I #30 Dec 01 2017 02:57:21
%S 1,4,30,280,2890,31584,358176,4168560,49455450,595480600,7254787540,
%T 89234708160,1106335812400,13808393670400,173332340911200,
%U 2186551157230560,27701981424940890,352297514508697800,4495418315974868700,57535568476437651600,738373616359119126540
%N Diagonal of the rational function 1/(1 - x - y - z + x*y + x*z - y*z).
%C Annihilating differential operator: (-2*x+29*x^2-27*x^3)*Dx^2 + (-2+58*x-81*x^2)*Dx + 8-24*x.
%H Gheorghe Coserea, <a href="/A274665/b274665.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],3456*x^5*(1-31/2*x+28*x^2-27/2*x^3)/(1-16*x+40*x^2)^3)/(1-16*x+40*x^2)^(1/4).
%F 0 = (-2*x+29*x^2-27*x^3)*y'' + (-2+58*x-81*x^2)*y' + (8-24*x)*y, where y is the g.f.
%F Recurrence: 2*n^2*a(n) = (29*n^2 - 29*n + 8)*a(n-1) - 3*(3*n - 4)*(3*n - 2)*a(n-2). - _Vaclav Kotesovec_, Jul 05 2016
%F a(n) ~ 3^(3*n + 3/2) / (Pi*sqrt(5)*n*2^(n+1)). - _Vaclav Kotesovec_, Jul 05 2016
%t a[0] = 1; a[1] = 4; a[n_] := a[n] = ((29*n^2 - 29*n + 8)*a[n-1] - 3*(3*n - 4)*(3*n - 2)*a[n-2])/(2*n^2);
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Dec 01 2017, after _Vaclav Kotesovec_ *)
%o (PARI)
%o my(x='x, y='y, z='z);
%o R = 1/(1 - x - y - z + x*y + x*z - 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 = 21; x = 'x + O('x^N);
%o Vec(hypergeom([1/12, 5/12],[1],3456*x^5*(1-31/2*x+28*x^2-27/2*x^3)/(1-16*x+40*x^2)^3, N)/(1-16*x+40*x^2)^(1/4))
%Y Cf. A268545-A268555.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Jul 01 2016