A276017 Diagonal of (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 8 x^2 y) * (1 - u - v - w)).

1, 18, 2160, 423360, 99792000, 25499650176, 6797581959168, 1860535606026240, 518890571236477440, 146835076503772800000, 42046646730013562757120, 12160617341681775057960960, 3547136319516173918512742400, 1042325945372157283978798694400, 308269259704090665806809006080000

Offset: 0

Author

Gheorghe Coserea, Aug 16 2016

Keywords

• nonn

Comments

"The corresponding (order-five) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(5,C)." (see A. Bostan link).

Examples

			1 + 18*x + 2160*x^2 + 423360*x^3 + ...
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..33
• 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, Eq. (C.3).
• Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Crossrefs

Cf. A004987, A268549, A268545-A268555.

Programs

Maple

diag_coeff := proc(expr, n)
    local var := [seq(indets(expr))], nvar := numelems(var);
    coeftayl(expr, var=[seq(0, i=1..nvar)], [seq(n, i=1..nvar)]);
end proc:
pxy := (1 - 3*y - 2*x + 3*y^2 + 8*x^2*y):
expr := (1 - 9*x*y)/(pxy * (1 - u - v - w)):
[seq(diag_coeff(expr, i), i=0..14)];

Mathematica

f = (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 8 x^2 y)*(1 - u - v - w));
a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}]&, f, {x, y, u, v, w}];
Array[a, 40, 0] (* Jean-François Alcover, Dec 03 2017 *)

Formula

a(n) = [(xyuvw)^n] (1 - 9*x*y)/((1 - 3*y - 2*x + 3*y^2 + 8*x^2*y) * (1 - u - v - w)).

From Vaclav Kotesovec, Dec 03 2017: (Start)

Recurrence: (n-1)^2*n^3*(3*n - 5)*a(n) = 18*(n-1)^2*(3*n - 4)*(3*n - 2)^2*(3*n - 1)*a(n-1) - 216*(3*n - 5)^2*(3*n - 4)*(3*n - 2)^2*(3*n - 1)*a(n-2).

a(n) ~ Gamma(1/3) * 2^(2*n - 10/3) * 3^(4*n + 1) / (Pi^2 * n^(4/3)). (End)