OFFSET
0,2
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).
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. (37).
FORMULA
a(n) = [(xyzuvw)^n] (1-9*x*y)/((1-3*y-2*x + 3*y^2 + 9*x^2*y) * (1-u-v-z-w)).
From Vaclav Kotesovec, Dec 03 2017: (Start)
Recurrence: n^4*a(n) = 24*(2*n - 1)*(3*n - 2)*(4*n - 3)*(4*n - 1)*a(n-1).
For n > 0, a(n) = 4 * 9^n * Gamma(4*n) * Gamma(n + 1/3) / (Gamma(1/3) * Gamma(n) * Gamma(n+1)^4).
a(n) ~ 2^(8*n - 1/2) * 3^(2*n) / (Pi^(3/2) * Gamma(1/3) * n^(13/6)). (End)
EXAMPLE
1 + 72*x + 45360*x^2 + 46569600*x^3 + ...
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:
expr := (1-9*x*y)/((1-3*y-2*x + 3*y^2 + 9*x^2*y) * (1-u-v-z-w)):
[seq(diag_coeff(expr, i), i=0..14)];
MATHEMATICA
f = (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y)*(1 - u - v - z - 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 *)
Join[{1}, Table[FullSimplify[(4 * 9^n * Gamma[4*n] * Gamma[1/3 + n]) / (Gamma[1/3] * Gamma[n] * Gamma[1 + n]^4)], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 03 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Aug 16 2016
STATUS
approved