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%I #38 Jul 31 2022 02:21:57
%S 1,12,648,50400,4630500,468087984,50345463168,5655718328832,
%T 656151696743400,78036148295820000,9465472643689782720,
%U 1166663950520357802240,145719568153188579382560,18405635030728188793200000
%N Diagonal of (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - z) * (1 - v - w)).
%C "The corresponding (order-three) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension." (see A. Bostan link) - _Gheorghe Coserea_, Aug 15 2016
%H Vaclav Kotesovec, <a href="/A268549/b268549.txt">Table of n, a(n) for n = 0..100</a>
%H A. Bostan, S. Boukraa, J.-M. Maillard, and 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. (30).
%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) = [(xyzuvw)^n] (1 - 9*x*y)/((1 - 3*y - 2*x + 3*y^2 + 9*x^2*y) * (1 - u - z) * (1 - v - w)).
%F D-finite with recurrence: n^3*a(n) -12*(3*n-2)*(-1+2*n)^2*a(n-1)=0. - _R. J. Mathar_, Mar 11 2016 [follows from the hypergeometric g.f. below - _Georg Fischer_, Jul 30 2022]
%F From _Vaclav Kotesovec_, Jul 01 2016: (Start)
%F a(n) = 3^(2*n) * (2*n)!^2 * Gamma(n + 1/3) / (Gamma(1/3) * (n!)^5).
%F a(n) ~ 12^(2*n)/(Gamma(1/3)*Pi*n^(5/3)).
%F (End)
%F From _Gheorghe Coserea_, Aug 16 2016: (Start)
%F a(n) = [(xyzuv)^n] 1/((1 - x + 3*y - 27*x*y^3 - 27*x*y^2 - 9*x*y + 3*y^2) * (1 - u - v - u*z - v*z)).
%F G.f.: hypergeom([1/3, 1/2, 1/2], [1, 1], 144*x).
%F (End)
%e 1 + 12*x + 648*x^2 + 50400*x^3 + ...
%p A268549 := proc(n)
%p (1-9*x*y)/(1-3*y-2*x+3*y^2+9*x^2*y)/(1-u-z)/(1-v-w) ;
%p coeftayl(%,x=0,n) ;
%p coeftayl(%,y=0,n) ;
%p coeftayl(%,z=0,n) ;
%p coeftayl(%,u=0,n) ;
%p coeftayl(%,v=0,n) ;
%p coeftayl(%,w=0,n) ;
%p end proc:
%p seq(A268549(n),n=0..40) ; # _R. J. Mathar_, Mar 11 2016
%p series(hypergeom([1/3, 1/2, 1/2], [1, 1], 144*x), x=0, 14); # _Gheorghe Coserea_, Aug 15 2016
%t FullSimplify[Table[3^(2*n)*(2*n)!^2*Gamma[n + 1/3]/(Gamma[1/3]*(n!)^5), {n, 0, 15}]] (* _Vaclav Kotesovec_, Jul 01 2016 *)
%Y Cf. A004987, A268545-A268555.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Feb 29 2016
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