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Search: id:A143397
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| A143397 |
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Triangle T(n,k)=number of forests of labeled rooted trees of height at most 1, with n labels and k nodes, where any root may contain >= 1 labels, n >= 0, 0<=k<=n. |
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2
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| 1, 0, 1, 0, 1, 3, 0, 1, 6, 10, 0, 1, 11, 36, 41, 0, 1, 20, 105, 230, 196, 0, 1, 37, 285, 955, 1560, 1057, 0, 1, 70, 756, 3535, 8680, 11277, 6322, 0, 1, 135, 2002, 12453, 41720, 80682, 86800, 41393, 0, 1, 264, 5347, 43008, 186669, 485982, 773724, 708948
(list; table; graph; listen)
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OFFSET
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0,6
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LINKS
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Index entries for sequences related to rooted trees
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FORMULA
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T(n, k) = Sum_{t=0..k} binomial(n,k-t) * stirling2(n-(k-t),t)*t^(k-t).
E.g.f.: exp(y*exp(x*y)*(exp(x)-1)). [From Vladeta Jovovic (vladeta(AT)eunet.yu), Dec 08 2008]
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EXAMPLE
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T (3,2) = 6: {1,2}{3}, {1,3}{2}, {2,3}{1}, {1,2}<-3, {1,3}<-2, {2,3}<-1.
Triangle begins:
[1]
[0, 1]
[0, 1, 3]
[0, 1, 6, 10]
[0, 1, 11, 36, 41]
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MAPLE
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with (combinat, stirling2): T := (n, k)-> add (binomial(n, k-t)* stirling2(n-(k-t), t)* t^(k-t), t=0..k); seq (seq (T(n, k), k=0..n), n=0..11);
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CROSSREFS
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Columns k=0-2: A000007, A000012, A006127. Diagonal: A000248. See also A048993, A008277, A007318, A143405 for row sums.
Sequence in context: A105147 A111924 A100485 this_sequence A137680 A161129 A011074
Adjacent sequences: A143394 A143395 A143396 this_sequence A143398 A143399 A143400
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 12 2008
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