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A276015
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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)).
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1
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1, 18, 1404, 158760, 21234150, 3126159036, 489778537248, 80153987120064, 13547671656870780, 2347445149320843000, 414851046001557525360, 74499573518808987538080, 13557818392046546526712440, 2495117936356342079352318000, 463604343771018075763879080000, 86854813070150110063356637257600
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OFFSET
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0,2
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COMMENTS
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"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).
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LINKS
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FORMULA
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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)).
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)
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EXAMPLE
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1 + 18*x + 1404*x^2 + 158760*x^3 + ...
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MAPLE
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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)];
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MATHEMATICA
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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}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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