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A143396
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Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, k of which are used for root nodes and any root may contain >= 1 labels, n >= 0, 0<=k<=n.
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2
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1, 0, 1, 0, 2, 2, 0, 3, 9, 5, 0, 4, 30, 40, 15, 0, 5, 90, 220, 185, 52, 0, 6, 255, 1040, 1485, 906, 203, 0, 7, 693, 4550, 9905, 9891, 4718, 877, 0, 8, 1820, 19040, 59850, 87416, 66808, 26104, 4140, 0, 9, 4644, 77448, 341082, 686826, 750120, 463212, 153063, 21147
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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LINKS
| Alois P. Heinz, Rows n = 0..99, flattened
Index entries for sequences related to rooted trees
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FORMULA
| T(n,k) = C(n,k) * Sum_{t=0..k} stirling2(k,t) * t^(n-k).
E.g.f.: exp(exp(x)*(exp(x*y)-1)). [From Vladeta Jovovic, Dec 08 2008]
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EXAMPLE
| T(3,2) = 9: {1,2}<-3, {1,3}<-2, {2,3}<-1, {1}<-3{2}, {1}{2}<-3, {1}<-2{3}, {1}{3}<-2, {2}<-1{3}, {2}{3}<-1.
Triangle begins:
1;
0, 1;
0, 2, 2;
0, 3, 9, 5;
0, 4, 30, 40, 15;
0, 5, 90, 220, 185, 52;
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MAPLE
| with (combinat): T:= (n, k)-> binomial(n, k) *add (stirling2 (k, t) *t^(n-k), t=0..k); seq (seq (T(n, k), k=0..n), n=0..11);
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CROSSREFS
| Columns k=0-1: A000007, A000027. Diagonal: A000110. See also A048993, A008277, A007318, A143405 for row sums.
Sequence in context: A118262 A065484 A011137 * A090657 A167001 A108563
Adjacent sequences: A143393 A143394 A143395 * A143397 A143398 A143399
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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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