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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.

%I A143397 #29 Aug 03 2021 15:37:17
%S A143397 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,
%T A143397 1560,1057,0,1,70,756,3535,8680,11277,6322,0,1,135,2002,12453,41720,
%U A143397 80682,86800,41393,0,1,264,5347,43008,186669,485982,773724,708948,293608
%N 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.
%H A143397 Alois P. Heinz, <a href="/A143397/b143397.txt">Rows n = 0..140, flattened</a>
%H A143397 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A143397 T(n,k) = Sum_{t=0..k} C(n,k-t) * Stirling2(n-(k-t),t) * t^(k-t).
%F A143397 E.g.f.: exp(y*exp(x*y)*(exp(x)-1)). - _Vladeta Jovovic_, Dec 08 2008
%e A143397 T(3,2) = 6: {1,2}{3}, {1,3}{2}, {2,3}{1}, {1,2}<-3, {1,3}<-2, {2,3}<-1.
%e A143397 Triangle begins:
%e A143397   1;
%e A143397   0, 1;
%e A143397   0, 1,  3;
%e A143397   0, 1,  6,  10;
%e A143397   0, 1, 11,  36,   41;
%e A143397   0, 1, 20, 105,  230,  196;
%e A143397   0, 1, 37, 285,  955, 1560,  1057;
%e A143397   0, 1, 70, 756, 3535, 8680, 11277, 6322;
%e A143397   ...
%p A143397 T:= (n,k)-> add(binomial(n, k-t)*Stirling2(n-(k-t),t)*t^(k-t), t=0..k):
%p A143397 seq(seq(T(n, k), k=0..n), n=0..11);
%t A143397 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;
%t A143397 Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 31 2016, translated from Maple *)
%Y A143397 Columns k=0-2: A000007, A000012, A006127. Diagonal: A000248. See also A048993, A008277, A007318, A143405 for row sums.
%K A143397 nonn,tabl
%O A143397 0,6
%A A143397 _Alois P. Heinz_, Aug 12 2008