A274781 Diagonal of the rational function 1/(1 - x - y - z - x y - x y z).

1, 9, 187, 4893, 141771, 4352799, 138747631, 4540248813, 151482515587, 5130182907699, 175813106080437, 6083513738182923, 212190873229751079, 7451115041129234211, 263154860370419749527, 9340227180994323327213, 332954350987408603124067, 11914280614112222340359211

Offset

0,2

Comments

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.

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..310

A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.

Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Formula

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).

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.

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

Mathematica

a[n_] := SeriesCoefficient[1/(1 - x - y - z - x y - x y z), {x, 0, n}, {y, 0, n}, {z, 0, n}];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Oct 23 2018 *)

Prog

(PARI)
my(x='x, y='y, z='z);
R = 1/(1 - x - y - z - x*y - x*y*z);
diag(n, expr, var) = {
  my(a = vector(n));
  for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
  for (k = 1, n, a[k] = expr;
       for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
  return(a);
};
diag(10, R, [x, y, z])
(PARI) system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 20; x = 'x + O('x^N);
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))

Crossrefs

Cf. A268545-A268555.

Sequence in context: A216409 A171194 A196297 * A293848 A351281 A266496

Adjacent sequences: A274778 A274779 A274780 * A274782 A274783 A274784

Keyword

nonn

Author

Gheorghe Coserea, Jul 06 2016

Status

approved