A268554 Diagonal of the rational function 1/((1 - w - u v) * (1 - x y - x z - y z)).

1, 36, 6300, 1552320, 445945500, 139815211536, 46384755633216, 16009450307136000, 5689533506261190300, 2067982222137781950000, 765185639177176836418800, 287266309673587605560908800, 109149488451384203661831720000

Offset

0,2

Comments

Each second element (which is zero) is skipped. - R. J. Mathar, Mar 10 2016

Annihilating differential operator: (-x^2+432*x^4)*Dx^4 + (-5*x+4320*x^3)*Dx^3 + (-4+10644*x^2)*Dx^2 + 6012*x*Dx + 288.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

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: n^3*(2*n-1)*a(n) -6*(4*n-1)*(3*n-1)*(3*n-2)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Mar 10 2016

From Vaclav Kotesovec, Jul 01 2016: (Start)

a(n) = (4*n)! * (3*n)! / ((n!)^3 * (2*n)!^2).

a(n) ~ 2^(4*n - 3/2) * 3^(3*n + 1/2) / (Pi^(3/2) * n^(3/2)).

(End)

0 = (-x^2+432*x^4)*y'''' + (-5*x+4320*x^3)*y''' + (-4+10644*x^2)*y'' + 6012*x*y' + 288*y, where y = 1 + 36*x^2 + 6300*x^4 + ... is the g.f. - Gheorghe Coserea, Jul 03 2016

Maple

A268554 := proc(n)
    1/(1-w-u*v)/(1-x*y-x*z-y*z) ;
    coeftayl(%, x=0, n) ;
    coeftayl(%, y=0, n) ;
    coeftayl(%, z=0, n) ;
    coeftayl(%, u=0, n) ;
    coeftayl(%, v=0, n) ;
    coeftayl(%, w=0, n) ;
end proc:
seq(A268554(2*n), n=0..40) ; # R. J. Mathar, Mar 10 2016

Mathematica

Table[(4*n)!*(3*n)!/((n!)^3*(2*n)!^2), {n, 0, 15}] (* Vaclav Kotesovec, Jul 01 2016 *)

Prog

(PARI)
my(x1='x1, x2='x2, x3='x3, y1='y1, y2='y2, y3='y3);
R = 1/((1 - y1 - y2*y3) * (1 - x1*x2 - x1*x3 - x2*x3));
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(11, R, [x1, x2, x3, y1, y2, y3]) \\ Gheorghe Coserea, Jun 30 2016

Crossrefs

Cf. A268545-A268555.

Sequence in context: A222336 A291975 A307351 * A326999 A068272 A068284

Adjacent sequences: A268551 A268552 A268553 * A268555 A268556 A268557

Keyword

nonn

Author

N. J. A. Sloane, Feb 29 2016

Status

approved