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A237195
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Number of simple labeled graphs on n nodes that contain some size k connected component, all of whose nodes are labeled with integers {1,2,...,k} for some k in {1,2,...,n}.
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1
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1, 2, 7, 52, 846, 28628, 1928768, 255610528, 66822534992, 34632302913632, 35711543058158592, 73426371674544520192, 301419451958411673103360, 2472252535617096234970201088, 40532629372281642451697543062528, 1328660058258732602631909956943781888
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OFFSET
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1,2
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COMMENTS
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In other words, a(n) is the number of simple labeled graphs on {1,2,...,n} such that 1 is an isolated node, or 1 and 2 form a size 2 component, or 1,2 and 3 form a size 3 component, or ... 1,2,3,...,k form a size k component, where 1<=k<=n.
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n} A223894(n,k)/binomial(n,k).
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EXAMPLE
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a(3) = 7. We count all 8 simple labeled graphs on {1,2,3} except: 1-3 2.
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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:
a:= n-> add(b(k)*2^((n-k)*(n-k-1)/2), k=1..n):
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MATHEMATICA
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nn=15; 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]; Map[Total, Table[Table[Drop[Transpose[Table[ Range[0, nn]!CoefficientList[Series[a[[n]]x^n/n! g, {x, 0, nn}], x], {n, 1, nn}]], 1][[i, j]]/Binomial[i, j], {j, 1, i}], {i, 1, nn}]]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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