A274666 Diagonal of the rational function 1/(1 - x - y + x y - x z - y z - x y z).

1, 5, 43, 461, 5491, 69395, 910855, 12274925, 168668035, 2352544535, 33204000853, 473179375355, 6797163712639, 98299113206663, 1429765398030943, 20899401842991341, 306819063154144675, 4521526749077118143, 66858281393757281641, 991598171159871109391

Offset

0,2

Comments

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.

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^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).

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.

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

a(n) ~ sqrt(53 + 191/sqrt(13)) * (47 + 13*sqrt(13))^n / (sqrt(2)*Pi*n*6^(n+1)). - Vaclav Kotesovec, Jul 05 2016

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A268545-A268555.

Sequence in context: A350117 A239265 A369023 * A301976 A083070 A191802

Adjacent sequences: A274663 A274664 A274665 * A274667 A274668 A274669

Keyword

nonn

Author

Gheorghe Coserea, Jul 02 2016

Status

approved