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A368597
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Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges.
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24
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1, 1, 3, 17, 150, 1803, 27364, 501015, 10736010, 263461265, 7283725704, 223967628066, 7581128184175, 280103206674480, 11216492736563655, 483875783716549277, 22371631078155742764, 1103548801569848115255, 57849356643299101021960, 3211439288584038922502820
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OFFSET
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0,3
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COMMENTS
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It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.
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LINKS
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Eric Weisstein's World of Mathematics, Graph Loop.
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FORMULA
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a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n). - Andrew Howroyd, Jan 06 2024
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EXAMPLE
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The a(0) = 1 through a(3) = 17 set-systems:
{} {{1}} {{1},{2}} {{1},{2},{3}}
{{1},{1,2}} {{1},{2},{1,3}}
{{2},{1,2}} {{1},{2},{2,3}}
{{1},{3},{1,2}}
{{1},{3},{2,3}}
{{2},{3},{1,2}}
{{2},{3},{1,3}}
{{1},{1,2},{1,3}}
{{1},{1,2},{2,3}}
{{1},{1,3},{2,3}}
{{2},{1,2},{1,3}}
{{2},{1,2},{2,3}}
{{2},{1,3},{2,3}}
{{3},{1,2},{1,3}}
{{3},{1,2},{2,3}}
{{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], {1, 2}], {n}], Union@@#==Range[n]&]], {n, 0, 5}]
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PROG
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(PARI) a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n, k) * binomial(binomial(k+1, 2), n)) \\ Andrew Howroyd, Jan 06 2024
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CROSSREFS
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This is the covering case of A014068.
Allowing edges of any positive size gives A054780, covering case of A136556.
The version contradicting strict AOC is A368730.
A000085 counts set partitions into singletons or pairs.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
Cf. A000272, A000666, A057500, A066383, A333331, A367869, A368596, A368600, A368928, A368951, A369199.
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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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