A143396 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.

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

Offset: 0

Author

Alois P. Heinz, Aug 12 2008

Keywords

• nonn
• tabl

Examples

			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;
  ...
		

Links

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

Crossrefs

Columns k=0-10 give: A000007, A000027, A273652, A273653, A273654, A273655, A273656, A273657, A273658, A273659, A273660.

Diagonal gives A000110.

Row sums give A143405.

T(2n,n) gives A273661.

Cf. A048993, A008277, A007318.

Programs

Maple

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);

Mathematica

T[n_, k_] := T[n, k] = Binomial[n, k]*Sum[StirlingS2[k, t]*If[n == k, 1, t^(n - k)], {t, 0, k}];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 27 2016, translated from Maple, updated Jan 01 2021 *)

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)). - Vladeta Jovovic, Dec 08 2008