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Search: id:A143395
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| A143395 |
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Triangle T(n,k)=number of forests of k labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, n >= 0, 0<=k<=n. |
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+0
8
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| 1, 0, 1, 0, 3, 1, 0, 7, 9, 1, 0, 15, 55, 18, 1, 0, 31, 285, 205, 30, 1, 0, 63, 1351, 1890, 545, 45, 1, 0, 127, 6069, 15421, 7770, 1190, 63, 1, 0, 255, 26335, 116298, 95781, 24150, 2282, 84, 1, 0, 511, 111645, 830845, 1071630, 416451, 62370, 3990, 108, 1, 0, 1023
(list; table; graph; listen)
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OFFSET
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0,5
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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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G.f. of column k: x^k/Prod_{t=k..2k}(1-tx). T(n, k) = Sum_{t=k..n} binomial(n,t)*stirling2(t,k)*k^(n-t).
E.g.f.: exp(y*exp(x)*(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) = 9: {1}{2}<-3, {1}{3}<-2, {1}{2,3}, {2}{1}<-3, {2}{3}<-1, {2}{1,3}, {3}{1}<-2, {3}{2}<-1, {3}{1,2}.
Triangle begins:
[1]
[0, 1]
[0, 3, 1]
[0, 7, 9, 1]
[0, 15, 55, 18, 1]
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MAPLE
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with (combinat, stirling2): T := (n, k)-> add (binomial(n, t)* stirling2(t, k)* k^(n-t), t=k..n); seq (seq (T(n, k), k=0..n), n=0..11);
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CROSSREFS
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Columns k=0-9: A000007, A000225, A016269(n-2), A028025(n-3), A143399, A143400, A143401, A143402, A143403, A143404. Diagonal: A000012. See also A048993, A008277, A007318, A143405 for row sums.
Sequence in context: A010601 A110504 A111246 this_sequence A090536 A052420 A103685
Adjacent sequences: A143392 A143393 A143394 this_sequence A143396 A143397 A143398
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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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