A274674 Diagonal of the rational function 1/(1 - x - x y - x z - y z + x y z).

1, 1, 7, 37, 211, 1351, 8611, 57037, 383587, 2615851, 18052057, 125693107, 882033439, 6229779739, 44246291467, 315774707437, 2263120500067, 16279948902259, 117498622706269, 850541100418807, 6173221388110861, 44912998208539561, 327476893004792197, 2392516335780421627

Offset

0,3

Comments

Annihilating differential operator: x*(6*x^2-4*x-5)*(2*x^4-64*x^3-27*x^2-3*x+1)*Dx^2 + (36*x^6-800*x^5+556*x^4+1496*x^3+411*x^2+30*x-5)*Dx + 12*x^5-100*x^4+256*x^3+540*x^2+105*x+5.

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

0 = x*(6*x^2-4*x-5)*(2*x^4-64*x^3-27*x^2-3*x+1)*y'' + (36*x^6-800*x^5+556*x^4+1496*x^3+411*x^2+30*x-5)*y' + (12*x^5-100*x^4+256*x^3+540*x^2+105*x+5)*y, where y is the g.f.

Recurrence: n^2*(517*n^2 - 2249*n + 2322)*a(n) = (1551*n^4 - 8298*n^3 + 13910*n^2 - 8103*n + 1530)*a(n-1) + (13959*n^4 - 88641*n^3 + 196637*n^2 - 178937*n + 54690)*a(n-2) + 2*(16544*n^4 - 121600*n^3 + 316309*n^2 - 336617*n + 117690)*a(n-3) - 2*(n-3)^2*(517*n^2 - 1215*n + 590)*a(n-4). - Vaclav Kotesovec, Jul 07 2016

Mathematica

gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*x^5*(1 - 3*x - 27*x^2 - 64*x^3 + 2*x^4)/(1 - 4*x - 18*x^2 - 28*x^3 + x^4)^3]/(1 - 4*x - 18*x^2 - 28*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 - 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 = 24; x = 'x + O('x^N);
Vec(hypergeom([1/12, 5/12], [1], 1728*x^5*(1-3*x-27*x^2-64*x^3+2*x^4)/(1-4*x-18*x^2-28*x^3+x^4)^3, N)/(1-4*x-18*x^2-28*x^3+x^4)^(1/4))

Crossrefs

Cf. A268545-A268555.

Sequence in context: A319013 A362247 A126475 * A255672 A077239 A362087

Adjacent sequences: A274671 A274672 A274673 * A274675 A274676 A274677

Keyword

nonn

Author

Gheorghe Coserea, Jul 06 2016

Status

approved