A274665 Diagonal of the rational function 1/(1 - x - y - z + x*y + x*z - y*z).

1, 4, 30, 280, 2890, 31584, 358176, 4168560, 49455450, 595480600, 7254787540, 89234708160, 1106335812400, 13808393670400, 173332340911200, 2186551157230560, 27701981424940890, 352297514508697800, 4495418315974868700, 57535568476437651600, 738373616359119126540

Offset

0,2

Comments

Annihilating differential operator: (-2*x+29*x^2-27*x^3)*Dx^2 + (-2+58*x-81*x^2)*Dx + 8-24*x.

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],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).

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.

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

a(n) ~ 3^(3*n + 3/2) / (Pi*sqrt(5)*n*2^(n+1)). - Vaclav Kotesovec, Jul 05 2016

Mathematica

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);
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 01 2017, after Vaclav Kotesovec *)

Prog

(PARI)
my(x='x, y='y, z='z);
R =  1/(1 - x - y - z + x*y + x*z - 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 = 21; x = 'x + O('x^N);
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))

Crossrefs

Cf. A268545-A268555.

Sequence in context: A370346 A378610 A375172 * A212073 A172392 A127130

Adjacent sequences: A274662 A274663 A274664 * A274666 A274667 A274668

Keyword

nonn

Author

Gheorghe Coserea, Jul 01 2016

Status

approved