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A106834 Triangle read by rows: T(n, m) = number of painted forests on labeled vertex set [n] with m trees. Also number of painted forests with exactly n - m edges.

%I A106834 #16 Feb 25 2015 05:29:08
%S A106834 1,1,2,3,6,3,16,30,18,4,125,220,135,40,5,1296,2160,1305,420,75,6,
%T A106834 16807,26754,15750,5180,1050,126,7,262144,401408,229824,75460,16100,
%U A106834 2268,196,8,4782969,7085880,3949722,1282176,278775,42336,4410,288,9
%N A106834 Triangle read by rows: T(n, m) = number of painted forests on labeled vertex set [n] with m trees. Also number of painted forests with exactly n - m edges.
%C A106834 Row sums equal A101313 (Number of painted forests - exactly one of its trees is painted - on labeled vertex set [n].).
%H A106834 Alois P. Heinz, <a href="/A106834/b106834.txt">Rows n = 1..141, flattened</a>
%H A106834 Washington Bomfim, <a href="http://webonfim.vilabol.uol.com.br/A106241.html">Illustration Of This Sequence.</a> [Broken link?]
%F A106834 T(n, m)= m * f(n, m), where f(n, m) = number of forests with n nodes and m labeled trees, A105599.
%F A106834 E.g.f.: y*B(x)*exp(y*B(x)), where B(x) is e.g.f. for A000272. - _Vladeta Jovovic_, May 24 2005
%e A106834 T(4,3) = 18 because there are 18 such forests with 4 nodes and 3 trees. (See the illustration of this sequence).
%e A106834 Triangle begins:
%e A106834 1;
%e A106834 1,         2;
%e A106834 3,         6,     3;
%e A106834 16,       30,    18,    4;
%e A106834 125,     220,   135,   40,    5;
%e A106834 1296,   2160,  1305,  420,   75,   6;
%e A106834 16807, 26754, 15750, 5180, 1050, 126,  7;
%p A106834 f:= proc(n,m) option remember;
%p A106834       if n<0 then 0
%p A106834     elif n=m then 1
%p A106834     elif m<1 or m>n then 0
%p A106834     else add(binomial(n-1,j-1) *j^(j-2) *f(n-j,m-1), j=1..n-m+1)
%p A106834       fi
%p A106834     end:
%p A106834 T:= (n,m)-> m*f(n,m):
%p A106834 seq(seq(T(n, m), m=1..n), n=1..12); # _Alois P. Heinz_, Sep 10 2008
%t A106834 f[n_, m_] := f[n, m] = Which[n<0, 0, n == m, 1, m<1 || m>n, 0, True, Sum[ Binomial[n-1, j-1]*j^(j-2)*f[n-j, m-1], {j, 1, n-m+1}]]; T[n_, m_] := m*f[n, m]; Table[Table[T[n, m], {m, 1, n}], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Feb 25 2015, after _Alois P. Heinz_ *)
%Y A106834 Cf. A101313, A105599, A106240.
%K A106834 easy,nonn,tabl
%O A106834 1,3
%A A106834 _Washington Bomfim_, May 19 2005