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A219034
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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.
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2
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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
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OFFSET
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0,4
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COMMENTS
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Row sums = (n+1)^(n-1).
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LINKS
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FORMULA
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E.g.f.: exp(T(x)-x+y*x) where T(x) is the e.g.f. for A000169.
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EXAMPLE
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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;
...
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MAPLE
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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)):
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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