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
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
Crossrefs
Programs
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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);
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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