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 A268549 Diagonal of (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - z) * (1 - v - w)). 6
 1, 12, 648, 50400, 4630500, 468087984, 50345463168, 5655718328832, 656151696743400, 78036148295820000, 9465472643689782720, 1166663950520357802240, 145719568153188579382560, 18405635030728188793200000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS "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 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..100 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. (30). FORMULA 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)). Conjecture: n^3*a(n) -12*(3*n-2)*(-1+2*n)^2*a(n-1)=0. - R. J. Mathar, Mar 11 2016 From Vaclav Kotesovec, Jul 01 2016: (Start) 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) From Gheorghe Coserea, Aug 16 2016: (Start) 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) EXAMPLE 1 + 12*x + 648*x^2 + 50400*x^3 + ... MAPLE A268549 := proc(n)     (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: seq(A268549(n), n=0..40) ; # R. J. Mathar, Mar 11 2016 series(hypergeom([1/3, 1/2, 1/2], [1, 1], 144*x), x=0, 14); # Gheorghe Coserea, Aug 15 2016 MATHEMATICA 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 *) CROSSREFS Cf. A004987, A268545-A268555. Sequence in context: A126159 A220992 A241227 * A071307 A195574 A195554 Adjacent sequences:  A268546 A268547 A268548 * A268550 A268551 A268552 KEYWORD nonn AUTHOR N. J. A. Sloane, Feb 29 2016 STATUS approved

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Last modified August 2 02:53 EDT 2021. Contains 346409 sequences. (Running on oeis4.)