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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
1, 18, 1404, 158760, 21234150, 3126159036, 489778537248, 80153987120064, 13547671656870780, 2347445149320843000, 414851046001557525360, 74499573518808987538080, 13557818392046546526712440, 2495117936356342079352318000, 463604343771018075763879080000, 86854813070150110063356637257600 (list; graph; refs; listen; history; text; internal format)
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
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).
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)
EXAMPLE
1 + 18*x + 1404*x^2 + 158760*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:
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 *)
CROSSREFS
Sequence in context: A292609 A275047 A160252 * A210823 A258336 A211310
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Aug 16 2016
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)