A218688
Number of ways to linearly arrange the trees over all forests on n labeled nodes.
Original entry on oeis.org
1, 1, 3, 15, 106, 975, 11106, 151501, 2415960, 44221869, 915826600, 21211128411, 544126606992, 15334985416075, 471495297242256, 15719617534811625, 565271886957356416, 21820620411482896089, 900398349688515500160, 39564926462522623540519, 1845034125763359894240000
Offset: 0
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T:= -LambertW(-x):
egf:= 1/(1-T+T^2/2):
a:= n-> n! * coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..30); # Alois P. Heinz, Nov 04 2012
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nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[1/(1-t+t^2/2),{x,0,nn}],x]
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A218688_vec(n,A=List(1))={until(#A>n,listput(A,sum(k=1,#A,binomial(#A,k)*k^(k-2)*A[#A-k+1])));Vec(A)} \\ M. F. Hasler, Jan 26 2020
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.
Original entry on oeis.org
1, 1, 2, 3, 6, 3, 16, 30, 18, 4, 125, 220, 135, 40, 5, 1296, 2160, 1305, 420, 75, 6, 16807, 26754, 15750, 5180, 1050, 126, 7, 262144, 401408, 229824, 75460, 16100, 2268, 196, 8, 4782969, 7085880, 3949722, 1282176, 278775, 42336, 4410, 288, 9
Offset: 1
T(4,3) = 18 because there are 18 such forests with 4 nodes and 3 trees. (See the illustration of this sequence).
Triangle begins:
1;
1, 2;
3, 6, 3;
16, 30, 18, 4;
125, 220, 135, 40, 5;
1296, 2160, 1305, 420, 75, 6;
16807, 26754, 15750, 5180, 1050, 126, 7;
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f:= proc(n,m) option remember;
if n<0 then 0
elif n=m then 1
elif m<1 or m>n then 0
else add(binomial(n-1,j-1) *j^(j-2) *f(n-j,m-1), j=1..n-m+1)
fi
end:
T:= (n,m)-> m*f(n,m):
seq(seq(T(n, m), m=1..n), n=1..12); # Alois P. Heinz, Sep 10 2008
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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 *)
A218691
Number of ways to paint some (possibly none or all) of the trees over all forests on n labeled nodes.
Original entry on oeis.org
1, 2, 6, 26, 156, 1242, 12616, 158034, 2372880, 41725106, 843126624, 19277549898, 492447987136, 13907344659210, 430397513894016, 14487404695687298, 527023721684738304, 20605894357093102434, 861761850029367846400, 38387125875316048363386, 1814541564588778500135936
Offset: 0
-
T:= -LambertW(-x):
egf:= exp(T-T^2/2)^2:
a:= n-> n! * coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..30); # Alois P. Heinz, Nov 04 2012
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nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Exp[(t-t^2/2)]^2,{x,0,nn}],x]
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