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Search: id:A094021
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| A094021 |
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Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k components (1<=k<=n). |
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1
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| 1, 1, 1, 3, 3, 1, 12, 14, 6, 1, 55, 75, 40, 10, 1, 273, 429, 275, 90, 15, 1, 1428, 2548, 1911, 770, 175, 21, 1, 7752, 15504, 13328, 6370, 1820, 308, 28, 1, 43263, 95931, 93024, 51408, 17640, 3822, 504, 36, 1, 246675, 600875, 648945, 406980, 162792, 42840
(list; table; graph; listen)
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OFFSET
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1,4
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COMMENT
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Column k=1 yields A001764; column k=2 yields A026004.
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REFERENCES
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P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.
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FORMULA
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T(n, k)=binomial(n, k-1)*binomial(3n-2k-1, n-k)/(2n-k). G.f. G=G(t, z) satisfies G^3+(t^3*z^2-t^2*z-3)G^2+(t^2*z+3)G-1=0.
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EXAMPLE
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T(3,2)=3 because, with A,B,C denoting the vertices of a triangle, we have the 2-component forests (A,BC), (B,CA) and (C,AB).
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MAPLE
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T:=proc(n, k) if k<=n then binomial(n, k-1)*binomial(3*n-2*k-1, n-k)/(2*n-k) else 0 fi end: seq(seq(T(n, k), k=1..n), n=1..11);
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CROSSREFS
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Cf. A001764, A026004.
Sequence in context: A120870 A010029 A143603 this_sequence A062746 A115193 A039797
Adjacent sequences: A094018 A094019 A094020 this_sequence A094022 A094023 A094024
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004
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