A274669 Diagonal of the rational function 1/(1 - x - y - z + x y - x z - y z).

1, 8, 138, 2960, 70090, 1756608, 45678864, 1219013664, 33162009210, 915589703600, 25578044554348, 721420319128704, 20509529725235824, 586986330979489280, 16895932626393943680, 488743896405192037440, 14198840150264907505050, 414069243091986225102480, 12115901803035178006468500

Offset

0,2

Comments

Annihilating differential operator: x*(x-2)*(11*x+12)*(27*x^2+92*x-3)*Dx^2 + (891*x^4+2132*x^3-689*x^2-4488*x+72)*Dx + 264*x^3+592*x^2+768*x-576.

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*(27*x^2+92*x-3)*(x-2)^2/(1-32*x+88*x^2)^3)/(1-32*x+88*x^2)^(1/4).

0 = x*(x-2)*(11*x+12)*(27*x^2+92*x-3)*y'' + (891*x^4+2132*x^3-689*x^2-4488*x+72)*y' + (264*x^3+592*x^2+768*x-576)*y, where y is the g.f.

D-finite with recurrence 72*(n^2)*a(n) +6*(-363*n^2+341*n-74)*a(n-1) +(-1601*n^2+7316*n-7460)*a(n-2) +2*(371*n^2-1531*n+1550)*a(n-3) +33*(3*n-8)*(3*n-10)*a(n-4)=0. - R. J. Mathar, Jul 27 2022

a(n) ~ sqrt(35/68 + 32/(17*sqrt(13))) * ((46 + 13*sqrt(13))/3)^n / (Pi*n). - Vaclav Kotesovec, Mar 19 2023

Mathematica

gf = Hypergeometric2F1[1/12, 5/12, 1, -1728*x^4*(27*x^2 + 92*x - 3)*(x - 2)^2/(1 - 32*x + 88*x^2)^3]/(1 - 32*x + 88*x^2)^(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*z - 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^4*(27*x^2+92*x-3)*(x-2)^2/(1-32*x+88*x^2)^3, N)/(1-32*x+88*x^2)^(1/4))

Crossrefs

Cf. A268545-A268555.

Sequence in context: A101388 A281684 A050789 * A187236 A301835 A091060

Adjacent sequences: A274666 A274667 A274668 * A274670 A274671 A274672

Keyword

nonn

Author

Gheorghe Coserea, Jul 05 2016

Status

approved