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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
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OFFSET
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0,6
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LINKS
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FORMULA
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T(n,k) = Sum_{t=0..k} C(n,k-t) * Stirling2(n-(k-t),t) * t^(k-t).
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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;
0, 1, 20, 105, 230, 196;
0, 1, 37, 285, 955, 1560, 1057;
0, 1, 70, 756, 3535, 8680, 11277, 6322;
...
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MAPLE
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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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MATHEMATICA
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T[n_, k_] := Sum[Binomial[n, k-t]*StirlingS2[n - (k-t), t]*t^(k-t), {t, 0, k}]; T[0, 0] = 1; T[_, 0] = 0;
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 31 2016, translated from Maple *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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