A274673 Diagonal of the rational function 1/(1 - x - y - z - x y + x y z).

1, 7, 127, 2851, 70651, 1853377, 50452459, 1409575699, 40147379587, 1160568048157, 33947097696337, 1002532535965429, 29843356238833879, 894349641410968477, 26955867982764111427, 816484373069154316051, 24838062486275592671587, 758470289246834941140037, 23239359305672548933204261

Offset

0,2

Comments

Annihilating differential operator: x*(5*x^2+4*x-6)*(x^4-3*x^3-27*x^2-64*x+2)*Dx^2 + (15*x^6-14*x^5-201*x^4-144*x^3+220*x^2+768*x-12)*Dx + 5*x^5+3*x^4+136*x^2+132*x+84.

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*(2-64*x-27*x^2-3*x^3+x^4)/(1-28*x-18*x^2-4*x^3+x^4)^3)/(1-28*x-18*x^2-4*x^3+x^4)^(1/4).

0 = x*(5*x^2+4*x-6)*(x^4-3*x^3-27*x^2-64*x+2)*y'' + (15*x^6-14*x^5-201*x^4-144*x^3+220*x^2+768*x-12)*y' + (5*x^5+3*x^4+136*x^2+132*x+84)*y, where y is the g.f.

Recurrence: 2*n^2*(517*n^2 - 1887*n + 1598)*a(n) = 2*(16544*n^4 - 76928*n^3 + 115285*n^2 - 64789*n + 11484)*a(n-1) + (13959*n^4 - 78867*n^3 + 152654*n^2 - 115150*n + 24984)*a(n-2) + (1551*n^4 - 10314*n^3 + 22982*n^2 - 18819*n + 3996)*a(n-3) - (n-3)^2*(517*n^2 - 853*n + 228)*a(n-4). - Vaclav Kotesovec, Jul 07 2016

Mathematica

gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*x^3*(2 - 64*x - 27*x^2 - 3*x^3 + x^4)/(1 - 28*x - 18*x^2 - 4*x^3 + x^4)^3]/(1 - 28*x - 18*x^2 - 4*x^3 + x^4)^(1/4);
CoefficientList[gf + O[x]^20, x] (* Jean-François Alcover, Dec 01 2017 *)

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*(2-64*x-27*x^2-3*x^3+x^4)/(1-28*x-18*x^2-4*x^3+x^4)^3, N)/(1-28*x-18*x^2-4*x^3+x^4)^(1/4))

Crossrefs

Cf. A268545-A268555.

Sequence in context: A025166 A241955 A139291 * A363870 A215066 A092676

Adjacent sequences: A274670 A274671 A274672 * A274674 A274675 A274676

Keyword

nonn

Author

Gheorghe Coserea, Jul 06 2016

Status

approved