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A052183
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a(n) = (n + 2) * binomial(3*n, n) / (2*n + 1).
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0
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2, 3, 12, 60, 330, 1911, 11424, 69768, 432630, 2713425, 17168580, 109390320, 700939512, 4512458580, 29164264320, 189120846288, 1229917589262, 8018580361365, 52392620853300, 342991368096300, 2249282417749290
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OFFSET
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0,1
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COMMENTS
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A second-order recursive sequence.
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LINKS
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FORMULA
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a(n) = (n+2)*c(2; n), where c(2; n) = binomial(3*n, n)/(2*n+1) (A001764).
c(2; n) is equivalent to Eq. (6.22) on p. 129 of the Carlitz reference.
a(n) = binomial(n+2, 2) * A0000139(n). - F. Chapoton, Feb 23 2024
G.f.: (2-5*g)/((3*g-1)*(g-1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011
Conjecture: 2*n*(2*n+1)*a(n) + (-47*n^2+50*n-12)*a(n-1) + 15*(3*n-4)*(3*n-5)*a(n-2) = 0. - R. J. Mathar, Sep 27 2012
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MATHEMATICA
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Table[(n+2) Binomial[3n, n]/(2n+1), {n, 0, 20}] (* Harvey P. Dale, Mar 23 2013 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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New name using a formula of the author by Peter Luschny, Feb 23 2024
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STATUS
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approved
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