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A219034
Triangular array read by rows: T(n,k) is the number of forests of rooted trees on n labeled nodes with exactly k isolated nodes; n>=0, 0<=k<=n.
2
1, 0, 1, 2, 0, 1, 9, 6, 0, 1, 76, 36, 12, 0, 1, 805, 380, 90, 20, 0, 1, 10626, 4830, 1140, 180, 30, 0, 1, 167839, 74382, 16905, 2660, 315, 42, 0, 1, 3091768, 1342712, 297528, 45080, 5320, 504, 56, 0, 1, 65127465, 27825912, 6042204, 892584, 101430, 9576, 756, 72, 0, 1
OFFSET
0,4
COMMENTS
Column k=0 is A105785.
Row sums = (n+1)^(n-1).
LINKS
FORMULA
E.g.f.: exp(T(x)-x+y*x) where T(x) is the e.g.f. for A000169.
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
2, 0, 1;
9, 6, 0, 1;
76, 36, 12, 0, 1;
805, 380, 90, 20, 0, 1;
10626, 4830, 1140, 180, 30, 0, 1;
167839, 74382, 16905, 2660, 315, 42, 0, 1;
3091768, 1342712, 297528, 45080, 5320, 504, 56, 0, 1;
...
MAPLE
b:= proc(n) option remember; expand(`if`(n=0, 1, add(i^(i-1)
*b(n-i)*binomial(n-1, i-1)*`if`(i=1, x, 1), i=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10); # Alois P. Heinz, Dec 31 2021
MATHEMATICA
nn=8; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0, nn]! CoefficientList[ Series[Exp[t-x+y x], {x, 0, nn}], {x, y}] //Grid
CROSSREFS
Sequence in context: A072551 A256117 A352372 * A372244 A256116 A185410
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Nov 10 2012
STATUS
approved