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%I #20 Dec 03 2017 08:17:32
%S 1,18,1620,211680,32744250,5578048476,1011486123648,191622065057280,
%T 37496545808513220,7522455177633825000,1539274868268143806800,
%U 320076654200479057190400,67449486609121784655852000,14374016718897406159444968000
%N Diagonal of (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - v - w)).
%C "The corresponding (order-three) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(3,C)." - (see A. Bostan link).
%H Gheorghe Coserea, <a href="/A276016/b276016.txt">Table of n, a(n) for n = 0..33</a>
%H A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015, Eq. (C.2).
%H Jacques-Arthur Weil, <a href="http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/">Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"</a>
%F a(n) = [(xyuvw)^n] (1-9*x*y)/((1-3*y-2*x+3*y^2+9*x^2*y) * (1-u-v-w)).
%F From _Vaclav Kotesovec_, Dec 03 2017: (Start)
%F Recurrence: n^3*a(n) = 9*(3*n - 2)^2*(3*n - 1)*a(n-1).
%F For n > 0, a(n) = 3^(2*n) * Gamma(3*n) * Gamma(n + 1/3) / (Gamma(4/3) * Gamma(n) * Gamma(n+1)^3).
%F a(n) ~ 3^(5*n + 1/2) / (2*Pi * Gamma(1/3) * n^(5/3)). (End)
%e 1 + 18*x + 1620*x^2 + 211680*x^3 + ...
%p diag_coeff := proc(expr, n)
%p local var := [seq(indets(expr))], nvar := numelems(var);
%p coeftayl(expr, var=[seq(0, i=1..nvar)], [seq(n, i=1..nvar)]);
%p end proc:
%p pxy := (1 - 3*y - 2*x + 3*y^2 + 9*x^2*y):
%p expr := (1 - 9*x*y)/(pxy * (1 - u - v - w)):
%p [seq(diag_coeff(expr, i), i=0..14)];
%t f = (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y)*(1 - u - v - w));
%t a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}]&, f, {x, y, u, v, w}];
%t Array[a, 40, 0] (* _Jean-François Alcover_, Dec 03 2017 *)
%t Join[{1}, Table[FullSimplify[3^(2*n) * Gamma[3*n] * Gamma[n + 1/3] / (Gamma[4/3] * Gamma[n] * Gamma[n + 1]^3)], {n, 1, 20}]] (* _Vaclav Kotesovec_, Dec 03 2017 *)
%Y Cf. A004987, A268549, A268545-A268555.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Aug 16 2016
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