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