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The unlabeled version is A368983 , connected case of A368984.

Exponential transform appears to be A333331. - Gus Wiseman, Feb 12 2024

ProofWiki, <a href="https://proofwiki.org/wiki/Definition:Loop-Graph">Definition:Loop-Graph</a>

Exponential transform appears to be A333331. - Gus Wiseman, Feb 12 2024

The case without loops is A057500, covering case of +A370317.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];

Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}]], Length[#]==Length[Union@@#]==n&&Length[csm[#]]<=1&]], {n, 0, 5}] (* Gus Wiseman, Feb 12 2024 *)

Cf. A000169, A057500, A333331, A063170, A053506.

The case of just pairs without loops is A057500, covering case of +.

Appears to be the connected case of A333331.

A058891 counts set-systems, unlabeled A000612.

A100861 counts set partitions into singletons or pairs by number of pairs.

A111924 counts set partitions into singletons or pairs by length.

Cf. A000272 labtrees, A000666 gra_loops, A054780 covs_vts_eq_eds, A116508 labgra_vts_eq_eds, A136556 setsys_n_eds_n_vts, A333331 loopgra_satis_aoc, A367863 labgra_cov_n_vts_n_eds, A367869 labgra_cov_satis_aoc, A368596 setsys_n_singpr_contra_aoc, A368599 unl_setsys_n_singpr_cov, A368600 setsys_n_n_contra_aoc.

Cf. A000272, A000666, A054780, A116508, A136556, A367863, A368600.

Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphLoop.html">Graph Loop</a>.

ProofWiki, <a href="https://proofwiki.org/wiki/Definition:Loop-Graph">Definition:Loop-Graph</a>

Exponential transform appears to be A333331. - Gus Wiseman, Feb 12 2024

From Gus Wiseman, Feb 12 2024: (Start)

The a(0) = 1 through a(3) = 10 loop-graphs:

{} {11} {11,12} {11,12,13}

{22,12} {11,12,23}

{11,13,23}

{22,12,13}

{22,12,23}

{22,13,23}

{33,12,13}

{33,12,23}

{33,13,23}

{12,13,23}

(End)

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];

Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}]], Length[#]==Length[Union@@#]==n&&Length[csm[#]]<=1&]], {n, 0, 5}] (* Gus Wiseman, Feb 12 2024 *)

This is the connected covering case of A014068.

The case of just pairs is A057500, covering case of +.

Allowing any number of edges gives A062740, connected case of A322661.

Appears to be the connected case of A333331.

This is the connected case of A368597.

The unlabeled version is A368983 connected case of A368984.

For at most n edges we have A369197.

A000085 counts set partitions into singletons or pairs.

A006129 counts covering graphs, connected A001187.

A058891 counts set-systems, unlabeled A000612.

A100861 counts set partitions into singletons or pairs by number of pairs.

A111924 counts set partitions into singletons or pairs by length.

Cf. A000272 labtrees, A000666 gra_loops, A054780 covs_vts_eq_eds, A116508 labgra_vts_eq_eds, A136556 setsys_n_eds_n_vts, A333331 loopgra_satis_aoc, A367863 labgra_cov_n_vts_n_eds, A367869 labgra_cov_satis_aoc, A368596 setsys_n_singpr_contra_aoc, A368599 unl_setsys_n_singpr_cov, A368600 setsys_n_n_contra_aoc.

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From Peter Luschny, Jan 10 2024: (Start)

a(n) = (exp(n)*Gamma(n + 1, n) - (n - 1)*n^(n - 1))/(2*n) for n > 0. - _Peter Luschny_, Jan 10 2024

a(n) = (1/2)*(A063170(n)/n - A053506(n)) for n > 0. (End)

Cf. A000169, A057500, A333331, A063170, A053506.

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