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A370319
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Triangle read by rows where T(n,k) is the number of labeled graphs with n vertices and k non-isolated vertices.
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0
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1, 1, 0, 1, 0, 1, 1, 0, 3, 4, 1, 0, 6, 16, 41, 1, 0, 10, 40, 205, 768, 1, 0, 15, 80, 615, 4608, 27449, 1, 0, 21, 140, 1435, 16128, 192143, 1887284, 1, 0, 28, 224, 2870, 43008, 768572, 15098272, 252522481, 1, 0, 36, 336, 5166, 96768, 2305716, 67942224, 2272702329, 66376424160
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OFFSET
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0,9
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LINKS
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FORMULA
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T(n,k) = binomial(n,k) * A006129(k).
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EXAMPLE
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Triangle begins:
1
1 0
1 0 1
1 0 3 4
1 0 6 16 41
1 0 10 40 205 768
1 0 15 80 615 4608 27449
Row n = 3 counts the following edge sets:
{} . {{1,2}} {{1,2},{1,3}}
{{1,3}} {{1,2},{2,3}}
{{2,3}} {{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
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MATHEMATICA
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Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Length[Union@@#]==k&]], {n, 0, 5}, {k, 0, n}]
Flatten@Table[Binomial[n, k]*Sum[(-1)^(k-m) Binomial[k, m] 2^Binomial[m, 2], {m, 0, k}], {n, 0, 10}, {k, 0, n}] (* Giorgos Kalogeropoulos, Feb 25 2024 *)
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CROSSREFS
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The unlabeled version is the partial subsequences of A002494.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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