A268551 Diagonal of 1/(1 - x + y + z + x y + x z - y z + x y z).

1, 11, 325, 11711, 465601, 19590491, 855266581, 38319499775, 1750193256961, 81131090245931, 3805404745303525, 180207832513958975, 8601942203526345025, 413358969518738106875, 19977566733574388828725, 970297391859524593324031, 47330511448436249282088961

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..170

A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227, 2015

Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Formula

Conjecture: n^2*(n-2)*a(n) +(-50*n^3+149*n^2-109*n+21)*a(n-1) -(2*n-3) *(51*n^2-153*n+91)*a(n-2) +(-50*n^3+301*n^2-565*n+315)*a(n-3) +(n-1)*(n-3)^2*a(n-4)=0. - R. J. Mathar, Mar 10 2016

a(n) ~ (1+sqrt(3))^(6*n+3) / (3*Pi*n*2^(3*n+3)). - Vaclav Kotesovec, Jul 01 2016

From Gheorghe Coserea, Jul 07 2016, (Start)

G.f.: hypergeom([1/12, 5/12],[1],13824*x^3*(x^2-52*x+1)/(x^2-46*x+1)^3/(x+1)^2)/((x^2-46*x+1)*(x+1)^2)^(1/4).

0 = x*(x-1)*(x^2-52*x+1)*(x+1)^2*y'' + (x+1)*(3*x^4-106*x^3+102*x^2+102*x-1)*y' + (x^4-12*x^3+32*x^2+68*x+11)*y, where y is g.f.

Annihilating differential operator: x*(x-1)*(x^2-52*x+1)*(x+1)^2*Dx^2 + (x+1)*(3*x^4-106*x^3+102*x^2+102*x-1)*Dx + x^4-12*x^3+32*x^2+68*x+11.

(End)

Maple

A268551 := proc(n)
    1/(1-x+y+z+x*y+x*z-y*z+x*y*z) ;
    coeftayl(%, x=0, n) ;
    coeftayl(%, y=0, n) ;
    coeftayl(%, z=0, n) ;
end proc:
seq(A268551(n), n=0..40) ; # R. J. Mathar, Mar 10 2016

Mathematica

gf = Hypergeometric2F1[1/12, 5/12, 1, 13824*x^3*(x^2 - 52*x + 1)/(x^2 - 46*x + 1)^3/(x + 1)^2]/((x^2 - 46*x + 1)*(x + 1)^2)^(1/4);
CoefficientList[gf + O[x]^40, x] (* Jean-François Alcover, Dec 03 2017, after Gheorghe Coserea *)

Prog

(PARI)
my(x='x, y='y, z='z);
R = 1/(1 - x + y + z + 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 = 20; x = 'x + O('x^N);
Vec(hypergeom([1/12, 5/12], [1], 13824*x^3*(x^2-52*x+1)/(x^2-46*x+1)^3/(x+1)^2, N)/((x^2-46*x+1)*(x+1)^2)^(1/4)) \\ Gheorghe Coserea, Jul 06 2016

Crossrefs

Cf. A268545-A268555.

Sequence in context: A270815 A197448 A241127 * A108274 A295171 A254545

Adjacent sequences: A268548 A268549 A268550 * A268552 A268553 A268554

Keyword

nonn

Author

N. J. A. Sloane, Feb 29 2016

Status

approved