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