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A268549
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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) * (1 - v - w)).
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6
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1, 12, 648, 50400, 4630500, 468087984, 50345463168, 5655718328832, 656151696743400, 78036148295820000, 9465472643689782720, 1166663950520357802240, 145719568153188579382560, 18405635030728188793200000
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OFFSET
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0,2
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COMMENTS
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"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
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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) * (1 - v - w)).
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]
a(n) = 3^(2*n) * (2*n)!^2 * Gamma(n + 1/3) / (Gamma(1/3) * (n!)^5).
a(n) ~ 12^(2*n)/(Gamma(1/3)*Pi*n^(5/3)).
(End)
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)).
G.f.: hypergeom([1/3, 1/2, 1/2], [1, 1], 144*x).
(End)
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EXAMPLE
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1 + 12*x + 648*x^2 + 50400*x^3 + ...
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MAPLE
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(1-9*x*y)/(1-3*y-2*x+3*y^2+9*x^2*y)/(1-u-z)/(1-v-w) ;
coeftayl(%, x=0, n) ;
coeftayl(%, y=0, n) ;
coeftayl(%, z=0, n) ;
coeftayl(%, u=0, n) ;
coeftayl(%, v=0, n) ;
coeftayl(%, w=0, n) ;
end proc:
series(hypergeom([1/3, 1/2, 1/2], [1, 1], 144*x), x=0, 14); # Gheorghe Coserea, Aug 15 2016
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MATHEMATICA
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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 *)
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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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