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%I #13 Oct 23 2018 12:51:59
%S 1,9,187,4893,141771,4352799,138747631,4540248813,151482515587,
%T 5130182907699,175813106080437,6083513738182923,212190873229751079,
%U 7451115041129234211,263154860370419749527,9340227180994323327213,332954350987408603124067,11914280614112222340359211
%N Diagonal of the rational function 1/(1 - x - y - z - x y - x y z).
%C Annihilating differential operator: x*(5*x^2-12*x-6)*(x^4-13*x^3+77*x^2-78*x+2)* Dx^2 + (15*x^6-178*x^5+823*x^4-1536*x^3-460*x^2+936*x-12)*Dx + 5*x^5-39*x^4+48*x^3+48*x^2-420*x+108.
%H Gheorghe Coserea, <a href="/A274781/b274781.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^3*(x^4-13*x^3+77*x^2-78*x+2)/(x^4-12*x^3+62*x^2-36*x+1)^3)/(x^4-12*x^3+62*x^2-36*x+1)^(1/4).
%F 0 = x*(5*x^2-12*x-6)*(x^4-13*x^3+77*x^2-78*x+2)*y'' + (15*x^6-178*x^5+823*x^4-1536*x^3-460*x^2+936*x-12)*y' + (5*x^5-39*x^4+48*x^3+48*x^2-420*x+108)*y, where y is g.f.
%F Recurrence: 2*n^2*(571*n^2 - 2169*n + 1898)*a(n) = 6*(7423*n^4 - 35620*n^3 + 54454*n^2 - 30721*n + 5364)*a(n-1) - (43967*n^4 - 254947*n^3 + 507958*n^2 - 395102*n + 87336)*a(n-2) + (7423*n^4 - 50466*n^3 + 117650*n^2 - 104391*n + 24732)*a(n-3) - (n-3)^2*(571*n^2 - 1027*n + 300)*a(n-4). - _Vaclav Kotesovec_, Jul 07 2016
%t a[n_] := SeriesCoefficient[1/(1 - x - y - z - x y - x y z), {x, 0, n}, {y, 0, n}, {z, 0, n}];
%t Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Oct 23 2018 *)
%o (PARI)
%o my(x='x, y='y, z='z);
%o R = 1/(1 - x - y - z - x*y - 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^3*(x^4-13*x^3+77*x^2-78*x+2)/(x^4-12*x^3+62*x^2-36*x+1)^3, N)/(x^4-12*x^3+62*x^2-36*x+1)^(1/4))
%Y Cf. A268545-A268555.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Jul 06 2016