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A033275
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Number of diagonal dissections of an n-gon into 3 regions.
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9
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0, 5, 21, 56, 120, 225, 385, 616, 936, 1365, 1925, 2640, 3536, 4641, 5985, 7600, 9520, 11781, 14421, 17480, 21000, 25025, 29601, 34776, 40600, 47125, 54405, 62496, 71456, 81345, 92225, 104160, 117216, 131461, 146965, 163800, 182040, 201761, 223041, 245960
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OFFSET
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4,2
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COMMENTS
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Number of standard tableaux of shape (n-3,2,2) (n>=5). - Emeric Deutsch, May 13 2004
Number of short bushes with n+1 edges and 3 branch nodes (i.e., nodes with outdegree at least 2). A short bush is an ordered tree with no nodes of outdegree 1. Example: a(5)=5 because the only short bushes with 6 edges and 3 branch nodes are the five full binary trees with 6 edges. Column 3 of A108263. - Emeric Deutsch, May 29 2005
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LINKS
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FORMULA
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a(n) = binomial(n+1, 2)*binomial(n-3, 2)/3.
Sum_{n>=5} 1/a(n) = 43/150.
Sum_{n>=5} (-1)^(n+1)/a(n) = 16*log(2)/5 - 154/75. (End)
E.g.f.: x*(exp(x)*(12 - 6*x + x^3) - 6*(2 + x))/12. - Stefano Spezia, Feb 21 2024
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MATHEMATICA
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a[4]=0; a[n_]:=Binomial[n+1, 2]*Binomial[n-3, 2]/3; Table[a[n], {n, 4, 43}] (* Indranil Ghosh, Feb 20 2017 *)
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PROG
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(PARI) concat(0, Vec(z^5*(5-4*z+z^2)/(1-z)^5 + O(z^60))) \\ Michel Marcus, Jun 18 2015
(Sage)
def A033275(n): return (binomial(n+1, 2)*binomial(n-3, 2))//3
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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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STATUS
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approved
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