A276013 Diagonal of (1 - 9 x y) / ((1 - 3 y - 2 x + 3 y^2 + 8 x^2 y) * (1 - u - z) * (1 - v - w)).

1, 12, 864, 100800, 14112000, 2139830784, 338341183488, 54913641209856, 9080061146956800, 1523231914913280000, 258557709598427086848, 44324863067728222027776, 7663322563977594870300672, 1334677098876385703362560000, 233951210561895726160281600000

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 + 12*x + 864*x^2 + 100800*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. (36).
• 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 - 9 x y)/((1 - 3y - 2x + 3 y^2 + 8 x^2 y)*(1 - 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 + 8*x^2*y) * (1-u-z) * (1-v-w)).

From Vaclav Kotesovec, Dec 03 2017: (Start)

Recurrence: (n-1)^2*n^3*(3*n - 5)*a(n) = 24*(n-1)^2*(2*n - 1)^2*(3*n - 4)*(3*n - 2)*a(n-1) - 384*(2*n - 3)^2*(2*n - 1)^2*(3*n - 5)*(3*n - 2)*a(n-2).

a(n) ~ Gamma(1/3) * 2^(6*n - 7/3) * 3^(n + 1/2) / (Pi^2 * n^(4/3)). (End)