A274667 Diagonal of the rational function 1/(1 - x - y - x y - x z - y z + x y z).

1, 3, 31, 339, 4131, 53013, 705139, 9618003, 133672387, 1884947073, 26889061761, 387207732453, 5619687743151, 82101265925409, 1206262382507451, 17809706204128659, 264074421220475427, 3930338612143125849, 58692717332813782501, 879093138034007102289, 13202346737893575996541

Offset

0,2

Comments

Annihilating differential operator: x*(2*x+1)*(6*x^2+x-8)*(x^3-41*x^2-29*x+2)*Dx^2 + (36*x^6-964*x^5-917*x^4+2394*x^3+2339*x^2+400*x-16)*Dx + 12*x^5-104*x^4+57*x^3+1067*x^2+640*x+48.

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^4*(x^3-41*x^2-29*x+2)*(1+2*x)^2/(1-12*x-34*x^2-36*x^3+x^4)^3)/(1-12*x-34*x^2-36*x^3+x^4)^(1/4).

0 = x*(2*x+1)*(6*x^2+x-8)*(x^3-41*x^2-29*x+2)*y'' + (36*x^6-964*x^5-917*x^4+2394*x^3+2339*x^2+400*x-16)*y' + (12*x^5-104*x^4+57*x^3+1067*x^2+640*x+48)*y, where y(x) is the g.f.

Recurrence: 2*n^2*(469*n^2 - 2106*n + 2229)*a(n) = (11725*n^4 - 64375*n^3 + 111011*n^2 - 68153*n + 13344)*a(n-1) + (46431*n^4 - 301356*n^3 + 678782*n^2 - 620403*n + 186048)*a(n-2) + (37989*n^4 - 284553*n^3 + 757682*n^2 - 829732*n + 299712)*a(n-3) - 2*(n-3)^2*(469*n^2 - 1168*n + 592)*a(n-4). - Vaclav Kotesovec, Jul 05 2016

Mathematica

CoefficientList[Series[HypergeometricPFQ[{1/12, 5/12}, {1}, 1728*x^4*(x^3-41*x^2-29*x+2)*(1+2*x)^2/(1-12*x-34*x^2-36*x^3+x^4)^3]/(1-12*x-34*x^2-36*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 = 22; x = 'x + O('x^N);
Vec(hypergeom([1/12, 5/12], [1], 1728*x^4*(x^3-41*x^2-29*x+2)*(1+2*x)^2/(1-12*x-34*x^2-36*x^3+x^4)^3, N)/(1-12*x-34*x^2-36*x^3+x^4)^(1/4))

Crossrefs

Cf. A268545-A268555.

Sequence in context: A152276 A136024 A051200 * A136596 A186207 A014178

Adjacent sequences: A274664 A274665 A274666 * A274668 A274669 A274670

Keyword

nonn

Author

Gheorghe Coserea, Jul 02 2016

Status

approved