A268542 The diagonal of the rational function 1/(1 - x - y - x y - x z - y z).

1, 4, 42, 520, 7090, 102144, 1525776, 23380368, 365130810, 5786380600, 92774019052, 1501646797248, 24498046138384, 402329384914240, 6645072333486720, 110293868867458080, 1838511122725436250, 30762545845461663240

Offset

0,2

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

Conjecture: 2*n^2*(21*n-37)*a(n) -32*(7*n-3)*(3*n^2-7*n+3)*a(n-1) +(-1281*n^3+4819*n^2-5610*n+1920)*a(n-2) -3*(3*n-5)*(21*n-16)*(3*n-7)*a(n-3) = 0. - R. J. Mathar, Mar 11 2016

G.f.: hypergeom([1/12, 5/12], [1], 1728*x^4*(x+1)^2*(27*x^2+34*x-2)/(-1+16*x+8*x^2)^3)/(1-16*x-8*x^2)^(1/4). - Gheorghe Coserea, Jul 06 2016

0 = x*(x+4)*(x+1)*(27*x^2+34*x-2)*y'' + (81*x^4+554*x^3+764*x^2+256*x-8)*y' + (24*x^3+184*x^2+192*x+32)*y, where y is g.f. - Gheorghe Coserea, Jul 06 2016

a(n) ~ sqrt(5/12 + 4/(3*sqrt(7))) * ((17+7*sqrt(7))/2)^n / (Pi*n). - Vaclav Kotesovec, Jul 07 2016

Maple

A268542 := proc(n)
    1/(1-x-y-x*y-x*z-y*z) ;
    coeftayl(%, x=0, n) ;
    coeftayl(%, y=0, n) ;
    coeftayl(%, z=0, n) ;
end proc:
seq(A268542(n), n=0..40) ; # R. J. Mathar, Mar 11 2016

Mathematica

gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*x^4*(x + 1)^2*(27*x^2 + 34*x - 2)/(-1 + 16*x + 8*x^2)^3]/(1 - 16*x - 8*x^2)^(1/4);
CoefficientList[gf + O[x]^18, x] (* Jean-François Alcover, Dec 02 2017, after Gheorghe Coserea *)

Prog

(PARI)
my(x='x, y='y, z='z);
R = 1/(1 - x - y - 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_sym([1/12, 5/12], [1], 1728*x^4*(x+1)^2*(27*x^2+34*x-2)/(-1+16*x+8*x^2)^3, N)/(1-16*x-8*x^2)^(1/4))  \\ Gheorghe Coserea, Jul 06 2016

Crossrefs

Cf. A268545-A268555.

Sequence in context: A335776 A092800 A370282 * A249928 A156440 A151453

Adjacent sequences: A268539 A268540 A268541 * A268543 A268544 A268545

Keyword

nonn

Author

N. J. A. Sloane, Feb 29 2016

Status

approved