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A228315
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Triangular array read by rows: T(n,k) is the number of rooted labeled simple graphs on {1,2,...,n} such that the root is in a component of size k; n>=1, 1<=k<=n.
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1
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1, 2, 2, 6, 6, 12, 32, 24, 48, 152, 320, 160, 240, 760, 3640, 6144, 1920, 1920, 4560, 21840, 160224, 229376, 43008, 26880, 42560, 152880, 1121568, 13063792, 16777216, 1835008, 688128, 680960, 1630720, 8972544, 104510336, 2012388736
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OFFSET
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1,2
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COMMENTS
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REFERENCES
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F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 7.
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LINKS
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FORMULA
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EXAMPLE
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1;
2, 2;
6, 6, 12;
32, 24, 48, 152;
320, 160, 240, 760, 3640;
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MAPLE
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b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
end:
T:= (n, k)-> binomial(n, k)*k*b(k)*2^((n-k)*(n-k-1)/2):
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MATHEMATICA
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nn = 10; g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; a =
Drop[Range[0, nn]! CoefficientList[Series[Log[g], {x, 0, nn}], x],
1]; Table[
Table[Binomial[n, k] k a[[k]] 2^Binomial[n - k, 2], {k, 1, n}], {n,
1, 7}] // 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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