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A224510
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Number of simple labeled graphs on {1,2,...,n} such that the node labeled with 1 is in the same component as the node labeled with 2.
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2
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0, 0, 1, 5, 48, 874, 30264, 2019680, 263757552, 68148453616, 35042313517056, 35957170070748800, 73714223732206510848, 302083108644327384484864, 2475273899774743284992553984, 40559859846438312840086623738880, 1329146799084147159829387611140308992
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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MAPLE
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b:= proc(n) b(n):= `if`(n=0, 1, 2^binomial(n, 2)-
add(binomial(n, k)*k*b(k)*2^binomial(n-k, 2), k=0..n-1)/n)
end:
a:= n-> add(binomial(n-2, k)*b(k+2)*2^binomial(n-k-2, 2), k=0..n-2):
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MATHEMATICA
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(* by brute force counting *) nn=10; g=Sum[2^Binomial[n, 2]x^n/n!, {n, 0, nn}]; a=Drop[Range[0, nn]!CoefficientList[Series[Log[g]+1, {x, 0, nn}], x], 1]; f[list_]:=Product[a[[i]], {i, list}]; Table[Total[Map[f, Map[Length, Select[SetPartitions[n], MemberQ[#[[1]], 2]&], {2}]]], {n, 2, nn}]
(* or *)
nn=30; g=Sum[2^Binomial[n, 2]x^n/n!, {n, 0, nn+2}]; Range[0, nn]!CoefficientList[Series[D[D[Log[g]+1, x], x] g, {x, 0, nn}], x]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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