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A372195
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Number of labeled simple graphs covering n vertices with a unique undirected cycle of length > 2.
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14
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0, 0, 0, 1, 15, 232, 3945, 75197, 1604974, 38122542, 1000354710, 28790664534, 902783451933, 30658102047787, 1121532291098765, 43985781899812395, 1841621373756094796, 82002075703514947236, 3869941339069299799884, 192976569550677042208068, 10139553075163838030949495
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OFFSET
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0,5
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COMMENTS
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An undirected cycle in a graph is a sequence of distinct vertices, up to rotation and reversal, such that there are edges between all consecutive elements, including the last and the first.
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LINKS
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FORMULA
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EXAMPLE
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The a(4) = 15 graphs:
12,13,14,23
12,13,14,24
12,13,14,34
12,13,23,24
12,13,23,34
12,13,24,34
12,14,23,24
12,14,23,34
12,14,24,34
12,23,24,34
13,14,23,24
13,14,23,34
13,14,24,34
13,23,24,34
14,23,24,34
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MATHEMATICA
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cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations/@Subsets[Union@@y, {k}], And@@Table[MemberQ[Sort/@y, Sort[{#[[i]], #[[If[i==k, 1, i+1]]]}]], {i, k}]&], {k, 3, Length[y]}], Min@@#==First[#]&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Length[cyc[#]]==2&]], {n, 0, 5}]
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PROG
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(PARI) seq(n)={my(w=lambertw(-x+O(x*x^n))); Vec(serlaplace(exp(-w-w^2/2-x)*(-log(1+w)/2 + w/2 - w^2/4)), -n-1)} \\ Andrew Howroyd, Jul 31 2024
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CROSSREFS
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A002807 counts cycles in a complete graph.
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KEYWORD
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nonn,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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