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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, 293608
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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LINKS
| Alois P. Heinz, Rows n = 0..140, flattened
Index entries for sequences related to rooted trees
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FORMULA
| T(n,k) = Sum_{t=0..k} C(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, Dec 08 2008]
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EXAMPLE
| 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
| with (combinat): 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
| Columns k=0-2: A000007, A000012, A006127. Diagonal: A000248. See also A048993, A008277, A007318, A143405 for row sums.
Sequence in context: A105147 A111924 A100485 * A137680 A201663 A199606
Adjacent sequences: A143394 A143395 A143396 * A143398 A143399 A143400
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KEYWORD
| nonn,tabl
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AUTHOR
| Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 12 2008
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