A322661 -id:A322661 - cp's OEIS frontend

A369147 Number of unlabeled loop-graphs covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

0, 0, 1, 7, 52, 411, 4440, 73886, 2128608, 111533208, 10812478194, 1945437194308, 650378721118910, 404749938336301313, 470163239887698682289, 1022592854829028310302180, 4177826139658552046624979658, 32163829440870460348768017832607, 468021728889827507080865185809438918

Offset: 0

Gus Wiseman, Jan 23 2024

nonn

			The a(0) = 0 through a(3) = 7 loop-graphs (loops shown as singletons):
  .  .  {{1},{2},{1,2}}  {{1},{2},{3},{1,2}}
                         {{1},{2},{1,2},{1,3}}
                         {{1},{2},{1,3},{2,3}}
                         {{1},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3}}
                         {{1},{2},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(3) = 7 unlabeled non-choosable loop-graphs.

Without the choice condition we have A322700, labeled A322661.
The complement for exactly n edges is A368984, labeled A333331 (maybe).
The labeled complement is A369140, covering case of A368927.
The labeled version is A369142, covering case of A369141.
This is the covering case of A369146.
The complement is counted by A369200, covering case of A369145.
Without loops we have A369202, covering case of A140637.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, labeled A006125 (shifted left).
A002494 counts unlabeled covering graphs, labeled A006129.
A007716 counts non-isomorphic multiset partitions, connected A007718.
Cf. A000088, A000612, A005703, A055621, A062740, A134964, A137917, A140638, A367868, A368835, A369199.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]}, {i,Length[p]}])],{p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n] && Length[Select[Tuples[#],UnsameQ@@#&]]==0&]]],{n,0,4}]

First differences of A369146.

a(n) = A322700(n) - A369200(n). - Andrew Howroyd, Feb 02 2024

a(6) onwards from Andrew Howroyd, Feb 02 2024

A322635 Number of regular graphs with loops on n labeled vertices.

Original entry on oeis.org

2, 4, 4, 24, 78, 1908, 23368, 1961200, 75942758, 25703384940, 4184912454930, 4462909435830552, 2245354417775573206, 10567193418810168583576, 24001585002447984453495392, 348615956932626441906675011568, 2412972383955442904868321667433106, 162906453913051798826796439651249753404

Offset: 1

Gus Wiseman, Dec 21 2018

nonn

A graph is regular if all vertices have the same degree. A loop adds 2 to the degree of its vertex.

Wikipedia, Regular graph
Gus Wiseman, The a(4) = 24 regular graphs with loops.
Gus Wiseman, The a(5) = 78 regular graphs with loops.

Row sums of A333158.

Cf. A000666, A054921, A059441, A295193, A299353, A306017, A306021, A319189, A319190, A319612, A322659, A322661.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Select[Tuples[Range[n],2],OrderedQ]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,2n}],{n,6}]

for(n=1, 10, print1(A322635(n), ", ")) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

a(11)-a(18) from Andrew Howroyd, Aug 28 2019

A339843 Number of distinct sorted degree sequences among all n-vertex half-loop-graphs without isolated vertices.

Original entry on oeis.org

1, 1, 3, 9, 29, 97, 336, 1188, 4275, 15579, 57358, 212908, 795657, 2990221, 11291665, 42814783, 162920417, 621885767, 2380348729

Offset: 0

Gus Wiseman, Dec 27 2020

In the covering case, these degree sequences, sorted in decreasing order, are the same thing as half-loop-graphical partitions (A321729). An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex.
The following are equivalent characteristics for any positive integer n:
(1) the prime indices of n can be partitioned into distinct singletons or strict pairs, i.e., into a set of half-loops or edges;
(2) n can be factored into distinct primes or squarefree semiprimes;
(3) the prime signature of n is half-loop-graphical.

			The a(0) = 1 through a(3) = 9 sorted degree sequences:
  ()  (1)  (1,1)  (1,1,1)
           (2,1)  (2,1,1)
           (2,2)  (2,2,1)
                  (2,2,2)
                  (3,1,1)
                  (3,2,1)
                  (3,2,2)
                  (3,3,2)
                  (3,3,3)
For example, the half-loop-graphs
  {{1},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3}}
both have degrees y = (3,2,2), so y is counted under a(3).

Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross references.
The version for simple graphs is A004251, covering: A095268.
The non-covering version (it allows isolated vertices) is A029889.
The same partitions counted by sum are conjectured to be A321729.
These graphs are counted by A006125 shifted left, covering: A322661.
The version for full loops is A339844, covering: A339845.
These graphs are ranked by A340018 and A340019.
A006125 counts labeled simple graphs, covering: A006129.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A339659 counts graphical partitions of 2n into k parts.
Cf. A062740, A096373, A167171, A320461, A320893, A320921, A320923, A338915, A339560, A339841, A339842.

Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Select[Subsets[Subsets[Range[n],{1,2}]],Union@@#==Range[n]&]]],{n,0,5}]

a(n) = A029889(n) - A029889(n-1) for n > 0. - Andrew Howroyd, Jan 10 2024

a(7)-a(18) added (using A029889) by Andrew Howroyd, Jan 10 2024

A368836 Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on up to n vertices with k loops and n-k non-loops.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 2, 6, 6, 2, 1, 6, 17, 18, 8, 2, 1, 21, 52, 58, 30, 9, 2, 1, 65, 173, 191, 107, 37, 9, 2, 1, 221, 585, 666, 393, 148, 39, 9, 2, 1, 771, 2064, 2383, 1493, 589, 168, 40, 9, 2, 1, 2769, 7520, 8847, 5765, 2418, 718, 176, 40, 9, 2, 1

Offset: 0

Gus Wiseman, Jan 11 2024

Are the row sums the same as column k = 1 (shifted left)?

Yes. When k = 1 there is one loop. Remove the vertex with the loop and add loops to its neighbors. This process is reversible so there is a bijection. - Andrew Howroyd, Jan 13 2024

			Triangle begins:
   1
   0  1
   0  1  1
   1  2  2  1
   2  6  6  2  1
   6 17 18  8  2  1
  21 52 58 30  9  2  1
Representatives of the loop-graphs counted by row n = 4:
  {12}{13}{14}{23} {1}{12}{13}{14} {1}{2}{12}{13} {1}{2}{3}{12} {1}{2}{3}{4}
  {12}{13}{24}{34} {1}{12}{13}{23} {1}{2}{12}{34} {1}{2}{3}{14}
                   {1}{12}{13}{24} {1}{2}{13}{14}
                   {1}{12}{23}{24} {1}{2}{13}{23}
                   {1}{12}{23}{34} {1}{2}{13}{24}
                   {1}{23}{24}{34} {1}{2}{13}{34}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Column k = 0 is A001434.
Row sums are A368598.
The labeled version is A368928.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts loop-graphs, unlabeled A000666.
A058891 counts set-systems, unlabeled A000612.
Cf. A000272, A007717, A062740, A322661, A333331, A368596, A368730, A368835.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}],{n}],Count[#,{_}]==k&]]], {n,0,4},{k,0,n}]

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
row(n) = {my(s=0, A=1+O(x*x^n)); forpart(p=n, s+=permcount(p) * polcoef(edges(p, i->A + x^i)*prod(i=1, #p, A + (x*y)^p[i]), n)); Vecrev(s/n!)} \\ Andrew Howroyd, Jan 13 2024

a(28) onwards from Andrew Howroyd, Jan 13 2024

A369200 Number of unlabeled loop-graphs covering n vertices such that it is possible to choose a different vertex from each edge (choosable).

Original entry on oeis.org

1, 1, 3, 7, 18, 43, 112, 282, 740, 1940, 5182, 13916, 37826, 103391, 284815, 788636, 2195414, 6137025, 17223354, 48495640, 136961527, 387819558, 1100757411, 3130895452, 8922294498, 25470279123, 72823983735, 208515456498, 597824919725, 1716072103910, 4931540188084

Offset: 0

Gus Wiseman, Jan 23 2024

nonn

These are covering loop-graphs with at most one cycle (unicyclic) in each connected component.

			Representatives of the a(1) = 1 through a(4) = 18 loop-graphs (loops shown as singletons):
  {{1}}  {{1,2}}      {{1},{2,3}}          {{1,2},{3,4}}
         {{1},{2}}    {{1,2},{1,3}}        {{1},{2},{3,4}}
         {{1},{1,2}}  {{1},{2},{3}}        {{1},{1,2},{3,4}}
                      {{1},{2},{1,3}}      {{1},{2,3},{2,4}}
                      {{1},{1,2},{1,3}}    {{1},{2},{3},{4}}
                      {{1},{1,2},{2,3}}    {{1,2},{1,3},{1,4}}
                      {{1,2},{1,3},{2,3}}  {{1,2},{1,3},{2,4}}
                                           {{1},{2},{3},{1,4}}
                                           {{1},{2},{1,3},{1,4}}
                                           {{1},{2},{1,3},{2,4}}
                                           {{1},{2},{1,3},{3,4}}
                                           {{1},{1,2},{1,3},{1,4}}
                                           {{1},{1,2},{1,3},{2,4}}
                                           {{1},{1,2},{2,3},{2,4}}
                                           {{1},{1,2},{2,3},{3,4}}
                                           {{1},{2,3},{2,4},{3,4}}
                                           {{1,2},{1,3},{1,4},{2,3}}
                                           {{1,2},{1,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Without the choice condition we have A322700, labeled A322661.
Without loops we have A368834, covering case of A134964.
For exactly n edges we have A368984, labeled A333331 (maybe).
The labeled version is A369140, covering case of A368927.
The labeled complement is A369142, covering case of A369141.
This is the covering case of A369145.
The complement is counted by A369147, covering case of A369146.
The complement without loops is A369202, covering case of A140637.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, labeled A006125 (shifted left).
A006129 counts covering graphs, unlabeled A002494.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A129271 counts connected choosable simple graphs, unlabeled A005703.
A133686 counts choosable labeled graphs, covering A367869.
Cf. A000088, A000612, A006649, A014068, A055621, A137916, A137917, A368835, A369194, A369199.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n]&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]]],{n,0,4}]

First differences of A369145.

Euler transform of A369289 with A369289(1) = 1. - Andrew Howroyd, Feb 02 2024

a(7) onwards from Andrew Howroyd, Feb 02 2024

A001862 Number of forests of least height with n nodes.

Original entry on oeis.org

1, 1, 2, 7, 26, 111, 562, 3151, 19252, 128449, 925226, 7125009, 58399156, 507222535, 4647051970, 44747776651, 451520086208, 4761032807937, 52332895618066, 598351410294193, 7102331902602676, 87365859333294151, 1111941946738682522, 14621347433458883187

Offset: 0

N. J. A. Sloane

nonn

From Gus Wiseman, Feb 14 2024: (Start)
Also the number of minimal loop-graphs covering n vertices. This is the minimal case of A322661. For example, the a(0) = 1 through a(3) = 7 loop-graphs are (loops represented as singletons):
  {}  {1}  {12}   {1-23}
           {1-2}  {2-13}
                  {3-12}
                  {12-13}
                  {12-23}
                  {13-23}
                  {1-2-3}
(End)

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983. See (3.3.7): number of ways to cover the complete graph K_n with star graphs.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
John Riordan, Letter to N. J. A. Sloane, Oct. 1970
John Riordan and N. J. A. Sloane, Correspondence, 1974

The connected case is A000272.
Without loops we have A053530, minimal case of A369191.
This is the minimal case of A322661.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 counts simple graphs; also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
Cf. A000085, A000169, A003465, A062740, A066383, A133686, A368597.

Range[0, 20]! CoefficientList[Series[Exp[x Exp[x] - x^2/2], {x, 0, 20}], x] (* Geoffrey Critzer, Mar 13 2011 *)
fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]& /@ Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length@fasmin[Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n]&]],{n,0,4}] (* Gus Wiseman, Feb 14 2024 *)

E.g.f.: exp(x*(exp(x)-x/2)).

Binomial transform of A053530. - Gus Wiseman, Feb 14 2024

Formula and more terms from Vladeta Jovovic, Mar 27 2001

A370168 Number of unlabeled loop-graphs with n vertices and at most n edges.

Original entry on oeis.org

1, 2, 5, 13, 36, 102, 313, 994, 3318, 11536, 41748, 156735, 609973, 2456235, 10224216, 43946245, 194866898, 890575047, 4190997666, 20289434813, 100952490046, 515758568587, 2703023502100, 14518677321040, 79852871813827, 449333028779385, 2584677513933282

Offset: 0

Gus Wiseman, Feb 16 2024

nonn

			The a(0) = 1 through a(3) = 13 loop-graph edge sets (loops shown as singletons):
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{1,2}}      {{1,2}}
             {{1},{2}}    {{1},{2}}
             {{1},{1,2}}  {{1},{1,2}}
                          {{1},{2,3}}
                          {{1,2},{1,3}}
                          {{1},{2},{3}}
                          {{1},{2},{1,2}}
                          {{1},{2},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The labeled version is A066383, covering A369194.
The case of equality is A368598, covering A368599.
The covering case is A370169, labeled A369194.
The loopless version is A370315, labeled A369192.
The covering loopless version is A370316, labeled A369191.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000666, A005703, A014068, A062740, A368597, A368927, A369141, A369196, A369197, A369199.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n], {1,2}]],Length[#]<=n&]]],{n,0,5}]

a(n)=my(A=O(x*x^n)); if(n==0, 1, polcoef(G(n, A)/(1-x), n)) \\ G defined in A070166. - Andrew Howroyd, Feb 19 2024

a(7) onwards from Andrew Howroyd, Feb 19 2024

A370316 Number of unlabeled simple graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 28, 68, 193, 534, 1568, 4635, 14146, 43610, 137015, 435227, 1400058, 4547768, 14917504, 49348612, 164596939, 553177992, 1872805144, 6385039022, 21917878860, 75739158828, 263438869515, 922219844982, 3249042441125, 11519128834499, 41097058489426

Offset: 0

Gus Wiseman, Feb 18 2024

nonn

			The a(0) = 1 through a(5) = 10 simple graphs:
  {}  .  {12}  {12-13}     {12-34}        {12-13-45}
               {12-13-23}  {12-13-14}     {12-13-14-15}
                           {12-13-24}     {12-13-14-25}
                           {12-13-14-23}  {12-13-23-45}
                           {12-13-24-34}  {12-13-24-35}
                                          {12-13-14-15-23}
                                          {12-13-14-23-25}
                                          {12-13-14-23-45}
                                          {12-13-14-25-35}
                                          {12-13-24-35-45}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The connected case is A005703, labeled A129271.
The case of exactly n edges is A006649, covering case of A001434.
The labeled version is A369191.
Partial row sums of A370167, covering case of A008406.
The non-covering version with loops is A370168, labeled A066383.
The version with loops is A370169, labeled A369194.
The non-covering version is A370315, labeled A369192.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A054548, A062734, A116508, A367862, A367863, A368599, A369193.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}],{0,n}], Union@@#==Range[n]&]]],{n,0,5}]

\\ G defined in A008406.
a(n)=my(A=O(x*x^n)); if(n==0, 1, polcoef((G(n,A)-G(n-1,A))/(1-x), n)) \\ Andrew Howroyd, Feb 19 2024

a(8) onwards from Andrew Howroyd, Feb 19 2024

A370315 Number of unlabeled simple graphs with n possibly isolated vertices and up to n edges.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 54, 146, 436, 1372, 4577, 15971, 58376, 221876, 876012, 3583099, 15159817, 66248609, 298678064, 1387677971, 6637246978, 32648574416, 165002122350, 855937433641, 4553114299140, 24813471826280, 138417885372373, 789683693019999, 4603838061688077

Offset: 0

Gus Wiseman, Feb 18 2024

nonn

			The a(1) = 1 through a(4) = 9 graph edge sets:
  {}  {}    {}          {}
      {12}  {12}        {12}
            {12-13}     {12-13}
            {12-13-23}  {12-34}
                        {12-13-14}
                        {12-13-23}
                        {12-13-24}
                        {12-13-14-23}
                        {12-13-24-34}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The case of exactly n edges is A001434, covering A006649.
The connected covering case is A005703, labeled A129271.
Partial row sums of A008406, covering A370167.
The labeled version is A369192.
The version with loops is A370168, labeled A066383.
The covering case is A370316, labeled A369191.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A000666, A322661, A322700, A369194, A369196, A369197, A369199, A370169.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]}, {i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}]], Length[#]<=n&]]],{n,0,5}]

a(n) = if(n<=1, n>=0, polcoef(G(n, O(x*x^n))/(1-x),n)) \\ G(n) defined in A008406. - Andrew Howroyd, Feb 20 2024

Sum of first n+1 terms of row n of A008406.

A368926 Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different element from each edge.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 1, 2, 5, 3, 1, 1, 5, 12, 7, 3, 1, 1, 14, 29, 19, 8, 3, 1, 1, 35, 75, 47, 21, 8, 3, 1, 1, 97, 191, 127, 54, 22, 8, 3, 1, 1, 264, 504, 331, 149, 56, 22, 8, 3, 1, 1, 733, 1339, 895, 395, 156, 57, 22, 8, 3, 1, 1

Offset: 0

Gus Wiseman, Jan 13 2024

Also the number of unlabeled loop-graphs covering n vertices with k loops and n-k non-loops such that each connected component has the same number of edges as vertices.

			Triangle begins:
   1
   0  1
   0  1  1
   1  2  1  1
   2  5  3  1  1
   5 12  7  3  1  1
  14 29 19  8  3  1  1
  35 75 47 21  8  3  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

The case of a unique choice is A106234, row sums A000081.
Column k = 0 is A137917, labeled version A137916.
Without the choice condition we have A368836.
The labeled version is A368924, row sums maybe A333331.
Row sums are A368984, complement A368835.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts loop-graphs, unlabeled A000666.
A322661 counts labeled covering half-loop-graphs, connected A062740.
Cf. A057500, A116508, A133686, A367863, A367869, A368596, A368597, A368598, A368601, A368836, A368927.

Table[Length[Union[sysnorm /@ Select[Subsets[Subsets[Range[n],{1,2}],{n}],Count[#,{_}]==k && Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]], {n,0,5},{k,0,n}]

\\ TreeGf gives gf of A000081; G(n,1) is gf of A368983.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
G(n,y)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); 1 + (sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2 - (g(1)^2 + g(2)))/2 + (y-1)*g(1)}
EulerMTS(p)={my(n=serprec(p,x)-1,vars=variables(p)); exp(sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i))}
T(n)={[Vecrev(p) | p <- Vec(EulerMTS(G(n,y) - 1))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2024

a(36) onwards from Andrew Howroyd, Jan 14 2024

cp's OEIS Frontend

A369147 Number of unlabeled loop-graphs covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

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A322635 Number of regular graphs with loops on n labeled vertices.

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A339843 Number of distinct sorted degree sequences among all n-vertex half-loop-graphs without isolated vertices.

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A368836 Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on up to n vertices with k loops and n-k non-loops.

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A369200 Number of unlabeled loop-graphs covering n vertices such that it is possible to choose a different vertex from each edge (choosable).

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A001862 Number of forests of least height with n nodes.

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A370168 Number of unlabeled loop-graphs with n vertices and at most n edges.

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