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A003408
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a(n) = binomial(3n+6, n).
(Formerly M4643)
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10
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1, 9, 66, 455, 3060, 20349, 134596, 888030, 5852925, 38567100, 254186856, 1676056044, 11058116888, 73006209045, 482320623240, 3188675231420, 21094923659355, 139646485582065, 925029565741050, 6131164307078475, 40661170824914640, 269807672771096460
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OFFSET
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0,2
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COMMENTS
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Number of connected graphs without crossing edges on n+3 nodes on a circle and having exactly 1 interior face. - Emeric Deutsch, Nov 06 2001
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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2*n*(n+3)*(2*n+5)*a(n) - 3*(3*n+5)*(3*n+4)*(n+2)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013
From Karol A. Penson, Feb 28 2024. (Start)
O.g.f.(z) = hypergeometric3F2([7/3, 8/3, 3], [7/2, 4], (27*z)/4).
O.g.f.(z) = g satisfies the algebraic equation: 1 + (-15*z^2+9*z-1)*g + (27*z-4)*z^3*g^2 + (27*z-4)*z^6*g^3 = 0.
a(n) is not a positive definite sequence, i.e. it cannot be represented as the n-th power moment of a positive weight function. (End)
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EXAMPLE
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a(0)=1 because among the 4 non-crossing connected graphs on 3 nodes on a circle only the triangle has exactly 1 interior face.
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MAPLE
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a:=n->sum(binomial(2*n-2, n+j)*binomial(n-1, n-j), j=0..n): seq(a(n), n=3..22); # Zerinvary Lajos, Jan 29 2007
R := RootOf(x-t*(t-1)^2, t); ogf := series(1/((1-3*R)*(1-R)^6), x=0, 20); # Mark van Hoeij, Nov 08 2011
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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