A276015 Diagonal of (1 - 9 x y) / ((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - z - u z) * (1 - v - w)).

1, 18, 1404, 158760, 21234150, 3126159036, 489778537248, 80153987120064, 13547671656870780, 2347445149320843000, 414851046001557525360, 74499573518808987538080, 13557818392046546526712440, 2495117936356342079352318000, 463604343771018075763879080000, 86854813070150110063356637257600

Offset: 0

Author

Gheorghe Coserea, Aug 16 2016

Keywords

• nonn

Comments

"The corresponding (order-four) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(4,C)." (see A. Bostan link).

Examples

			1 + 18*x + 1404*x^2 + 158760*x^3 + ...
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..22
• 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.1).
• 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 + 9*x^2*y):
expr := (1 - 9*x*y)/(pxy * (1-u-z-u*z) * (1-v-w)):
[seq(diag_coeff(expr, i), i=0..14)];

Mathematica

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

Formula

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

From Vaclav Kotesovec, Dec 03 2017: (Start)

Recurrence: (n-1)*n^3*a(n) = 18*(n-1)*(2*n - 1)^2*(3*n - 2)*a(n-1) - 36*(2*n - 3)*(2*n - 1)*(3*n - 5)*(3*n - 2)*a(n-2).

a(n) ~ Pi * 2^(2*n - 5/4) * 3^(2*n) * (1 + sqrt(2))^(2*n + 1) / (Gamma(1/3) * Pi^2 * n^(5/3)). (End)