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A129869
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Number of positive clusters of type D_n.
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1
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-1, 1, 5, 20, 77, 294, 1122, 4290, 16445, 63206, 243542, 940576, 3640210, 14115100, 54826020, 213286590, 830905245, 3241119750, 12657425550, 49483369320, 193641552390, 758454277620, 2973183318300, 11664026864100, 45791597230002, 179892016853724
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| This is also the number of antichains in the poset of positive-but-not-simple roots of type Dn.
If Y is a fixed 2-subset of a (2n+1)-set X then a(n+1) is the number of (n+2)-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Oct 28 2007
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REFERENCES
| F. Chapoton and L. Manivel, Triangulations and Severi varieties, Arxiv preprint arXiv:1109.6490, 2011
S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
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LINKS
| Milan Janjic, Two Enumerative Functions
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FORMULA
| a(n) = (3*n-4)/n * C(2*n-3,n-1).
Starting with "1" = the Narayana transform (A001263) of [1, 4, 7, 10, 13, 16,...]. - Gary W. Adamson, Jul 29 2001
G.f.: x^2*(sqrt(1-4*x)*(2*x+1)-4*x+1)/(sqrt(1-4*x)*(4*x^2-5*x+1) +12*x^2-7*x+1)-x. [From Vladimir Kruchinin, Sep 27 2011]
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EXAMPLE
| a(3) = 5 because the type D3 is the same as type A3 and there are 5 positive clusters among the 14 clusters in type A3.
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PROG
| (3*n-4)/n*binomial(2*n-3, n-1) $n=1..22; (MuPAD code)
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CROSSREFS
| Cf. A051924.
Sequence in context: A000758 A005283 A057552 * A079737 A028814 A079820
Adjacent sequences: A129866 A129867 A129868 * A129870 A129871 A129872
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KEYWORD
| sign
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AUTHOR
| F. Chapoton (fchapoton(AT)voila.fr), May 24 2007
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