A143397 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.

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

Offset: 0

Author

Alois P. Heinz, Aug 12 2008

Keywords

• nonn
• tabl

Examples

			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;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Index entries for sequences related to rooted trees

Crossrefs

Columns k=0-2: A000007, A000012, A006127. Diagonal: A000248. See also A048993, A008277, A007318, A143405 for row sums.

Programs

Maple

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);

Mathematica

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 *)

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)). - Vladeta Jovovic, Dec 08 2008