A274672 Diagonal of the rational function 1 / (1 - x - y - z - x y - y z + x y z).

1, 9, 211, 6093, 194211, 6552759, 229425631, 8244126669, 302001325987, 11228376095379, 422418832551981, 16044756006730779, 614293828071090831, 23677157141279202843, 917844665558530089591, 35756691970868828784813, 1399009670335978828299747, 54945874078445147692928043

Offset

0,2

Comments

Annihilating differential operator: x*(x+2)*(8*x^2-x-6)*(2*x^3-29*x^2-41*x+1)*Dx^2 + (48*x^6-408*x^5-777*x^4+798*x^3+1847*x^2+972*x-12)*Dx + 16*x^5-52*x^4+141*x^3+529*x^2+408*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*(2*x^3-29*x^2-41*x+1)*(x+2)^2/(1-36*x-34*x^2-12*x^3+x^4)^3)/(1-36*x-34*x^2-12*x^3+x^4)^(1/4).

0 = x*(x+2)*(8*x^2-x-6)*(2*x^3-29*x^2-41*x+1)*y'' + (48*x^6-408*x^5-777*x^4+798*x^3+1847*x^2+972*x-12)*y' + (16*x^5-52*x^4+141*x^3+529*x^2+408*x+108)*y, where y is the g.f.

D-finite with recurrence 12*(n^2)*a(n) +4*(-121*n^2+120*n-26)*a(n-1) +(-691*n^2+1608*n-860)*a(n-2) +(399*n^2-3591*n+6653)*a(n-3) +(779*n^2-6234*n+12331)*a(n-4) +2*(101*n^2-907*n+2036)*a(n-5) -16*(n-5)^2*a(n-6)=0. - R. J. Mathar, Jul 27 2022

Mathematica

gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*x^3*(2*x^3 - 29*x^2 - 41*x + 1)*(x + 2)^2/(1 - 36*x - 34*x^2 - 12*x^3 + x^4)^3]/(1 - 36*x - 34*x^2 - 12*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 - 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^3*(2*x^3-29*x^2-41*x+1)*(x+2)^2/(1-36*x-34*x^2-12*x^3+x^4)^3, N)/(1-36*x-34*x^2-12*x^3+x^4)^(1/4))

Crossrefs

Cf. A268545-A268555.

Sequence in context: A218887 A001535 A300136 * A122399 A320096 A327198

Adjacent sequences: A274669 A274670 A274671 * A274673 A274674 A274675

Keyword

nonn

Author

Gheorghe Coserea, Jul 05 2016

Status

approved