A116508 -id:A116508 - cp's OEIS frontend

A368951 Number of connected labeled graphs with n edges and n vertices and with loops allowed.

1, 1, 2, 10, 79, 847, 11436, 185944, 3533720, 76826061, 1880107840, 51139278646, 1530376944768, 49965900317755, 1767387701671424, 67325805434672100, 2747849045156064256, 119626103584870552921, 5533218319763109888000, 270982462739224265922466

Offset: 0

Andrew Howroyd, Jan 10 2024

nonn

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)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A000169, A333331, A063170, A053506.
This is the connected covering case of A014068.
The case without loops is A057500, covering case of A370317.
Allowing any number of edges gives A062740, connected case of A322661.
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.
Cf. A000272, A000666, A054780, A116508, A136556, A367863, A368600.

egf:= (L-> 1-L/2-log(1+L)/2-L^2/4)(LambertW(-x)):
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 10 2024

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(-log(1-t)/2 + t/2 - t^2/4 + 1))}

a(n) = A000169(n) + A057500(n) for n > 0.
E.g.f.: 1 - log(1-T(x))/2 + T(x)/2 - T(x)^2/4 where T(x) = -LambertW(-x) is the e.g.f. of A000169.
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.
a(n) = (1/2)*(A063170(n)/n - A053506(n)) for n > 0.  (End)

A368730 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, such that it is not possible to choose a different element from each.

Original entry on oeis.org

0, 0, 0, 0, 6, 180, 4560, 117600, 3234588, 96119982, 3092585310, 107542211535, 4029055302855, 162040513972623, 6970457656110039, 319598974394563500, 15568332397812799920, 803271954062642638830, 43778508937914677872788, 2513783434620146896920843

Offset: 0

Gus Wiseman, Jan 04 2024

nonn

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The a(4) = 6 set-systems:
  {{1},{2},{1,2},{3,4}}
  {{1},{3},{1,3},{2,4}}
  {{1},{4},{1,4},{2,3}}
  {{2},{3},{1,4},{2,3}}
  {{2},{4},{1,3},{2,4}}
  {{3},{4},{1,2},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Graph Loop.
ProofWiki, Definition:Loop-Graph

The case of a unique choice appears to be A000272.
The version without the choice condition is A368597, non-covering A014068.
The complement appears to be A333331.
The non-covering case is A368596, allowing edges of any size A368600.
Allowing any number of edges of any size gives A367903, ranks A367907.
Allowing any number of non-singletons gives A367868, non-covering A367867.
A000085 counts set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
A322661 counts labeled covering half-loop-graphs, connected A062740.
Cf. A000666, A116508, A133686, A367863, A367869, A367901, A368532.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}], {n}],Union@@#==Range[n] && Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]

a(n) = A368596(n) + A368597(n) - A014068(n). - Andrew Howroyd, Jan 10 2024

Terms a(7) and beyond from Andrew Howroyd, Jan 10 2024

A368924 Triangle read by rows where T(n,k) is the number of labeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different vertex from each edge.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 1, 9, 6, 1, 15, 68, 48, 12, 1, 222, 720, 510, 150, 20, 1, 3670, 9738, 6825, 2180, 360, 30, 1, 68820, 159628, 110334, 36960, 6895, 735, 42, 1, 1456875, 3067320, 2090760, 721560, 145530, 17976, 1344, 56, 1, 34506640, 67512798, 45422928, 15989232, 3402756, 463680, 40908, 2268, 72, 1

Offset: 0

Gus Wiseman, Jan 10 2024

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Triangle begins:
      1
      0      1
      0      2      1
      1      9      6      1
     15     68     48     12      1
    222    720    510    150     20      1
   3670   9738   6825   2180    360     30      1
  68820 159628 110334  36960   6895    735     42      1
Row n = 3 counts the following loop-graphs:
  {{1,2},{1,3},{2,3}}  {{1},{1,2},{1,3}}  {{1},{2},{1,3}}  {{1},{2},{3}}
                       {{1},{1,2},{2,3}}  {{1},{2},{2,3}}
                       {{1},{1,3},{2,3}}  {{1},{3},{1,2}}
                       {{2},{1,2},{1,3}}  {{1},{3},{2,3}}
                       {{2},{1,2},{2,3}}  {{2},{3},{1,2}}
                       {{2},{1,3},{2,3}}  {{2},{3},{1,3}}
                       {{3},{1,2},{1,3}}
                       {{3},{1,2},{2,3}}
                       {{3},{1,3},{2,3}}

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

Column k = n-1 is A002378.
The case of a unique choice is A061356, row sums A000272.
Column k = 0 is A137916, unlabeled version A137917.
Row sums appear to be A333331.
The complement has row sums A368596, covering case A368730.
The unlabeled version is A368926.
Without the choice condition we have A368928, A116508, A367863, A368597.
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.
Cf. A000169, A057500, A062740, A129271, A133686, A322661, A367869, A367902, A368601, A368835, A368836, A368927.

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

T(n)={my(t=-lambertw(-x + O(x*x^n))); [Vecrev(p) | p <- Vec(serlaplace(exp(-log(1-t)/2 - t/2 + t*y - t^2/4)))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 14 2024

E.g.f.: A(x,y) = exp(-log(1-T(x))/2 - T(x)/2 + y*T(x) - T(x)^2/4) where T(x) = -LambertW(-x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 14 2024

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

A367917 BII-numbers of set-systems with the same number of edges as covered vertices.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 17, 19, 21, 22, 24, 26, 28, 34, 35, 37, 38, 40, 41, 44, 49, 50, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 128, 129, 130, 131, 133, 134, 136, 137, 138, 139, 141, 142, 145, 147, 149, 150, 152

Offset: 1

Gus Wiseman, Dec 12 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

			The terms together with the corresponding set-systems begin:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  26: {{2},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  34: {{2},{2,3}}
  35: {{1},{2},{2,3}}
  37: {{1},{1,2},{2,3}}

These set-systems are counted by A054780 and A367916, A368186.
Graphs of this type are A367862, covering A367863, unlabeled A006649.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, connected A323818, unlabeled A000612.
A070939 gives length of binary expansion.
A136556 counts set-systems on {1..n} with n edges.
Cf. A057500, A059201, A072639, A096111, A116508, A309326, A326031, A326702, A326753, A326754, A367770, A367902, A367905.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
Select[Range[0,100], Length[bpe[#]]==Length[Union@@bpe/@bpe[#]]&]

A369143 Number of labeled simple graphs with n edges and n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 0, 0, 0, 30, 1335, 47460, 1651230, 59636640, 2284113762, 93498908580, 4099070635935, 192365988161490, 9646654985111430, 515736895712230192, 29321225548502776980, 1768139644819077541440, 112805126206185257070660, 7595507651522103787077270, 538504704005397535690160274

Offset: 0

Gus Wiseman, Jan 21 2024

nonn

			The term a(5) = 30 counts all permutations of the graph {{1,2},{1,3},{1,4},{2,3},{2,4}}.

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version without the choice condition is A116508, covering A367863.
The complement is A137916.
Allowing any number of edges gives A367867, covering A367868.
The version with loops is A368596, covering A368730, unlabeled A368835.
For set-systems we have A368600, for any number of edges A367903.
The covering case is A369144.
A006125 counts simple graphs, unlabeled A000088.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
Cf. A000085, A057500, A062740, A129271, A133686, A333331, A367769, A367869, A367901, A367907.

Table[Length[Select[Subsets[Subsets[Range[n],{2}], {n}],Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]

a(n) = A116508(n) - A137916(n). - Andrew Howroyd, Feb 02 2024

a(8) onwards from Andrew Howroyd, Feb 02 2024

A369196 Number of labeled loop-graphs with n vertices and at most as many edges as covered vertices.

Original entry on oeis.org

1, 2, 7, 39, 320, 3584, 51405, 900947, 18661186, 445827942, 12062839691, 364451604095, 12157649050827, 443713171974080, 17583351295466338, 751745326170662049, 34485624653535808340, 1689485711682987916502, 88030098291829749593643, 4860631073631586486397141

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 1 through a(2) = 7 loop-graphs:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

The version counting all vertices is A066383, without loops A369192.
The loopless case is A369193, with case of equality A367862.
The covering case is A369194, connected A369197, minimal case A001862.
The case of equality is A369198, covering case A368597.
A000085, A100861, A111924 count set partitions into singletons or pairs.
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.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, A000666, A001187, A006649, A057500, A062740, A116508, A367863, A369145.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]],Length[#]<=Length[Union@@#]&]],{n,0,5}]

Binomial transform of A369194.

A370169 Number of unlabeled loop-graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 1, 3, 7, 19, 48, 135, 373, 1085, 3184, 9590, 29258, 90833, 285352, 908006, 2919953, 9487330, 31111997, 102934602, 343389708, 1154684849, 3912345408, 13353796977, 45906197103, 158915480378, 553897148543, 1943627750652, 6865605601382, 24411508473314, 87364180212671, 314682145679491

Offset: 0

Gus Wiseman, Feb 16 2024

nonn

			The a(0) = 1 through a(4) = 19 loop-graph edge sets (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},{1,2},{3,4}}
                                               {{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..50

The case of equality is A368599, covering case of A368598.
The labeled version is A369194, covering case of A066383.
This is the covering case of A370168.
The loopless version is the covering case of A370315, labeled A369192.
This is the 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, A116508, 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}]], Union@@#==Range[n]&&Length[#]<=n&]]],{n,0,5}]

\\ G defined in A070166.
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(7) onwards from Andrew Howroyd, Feb 19 2024

A369193 Number of labeled simple graphs with n vertices and at most as many edges as covered (non-isolated) vertices.

Original entry on oeis.org

1, 1, 2, 8, 57, 608, 8614, 151365, 3162353, 76359554, 2088663444, 63760182536, 2147325661180, 79051734050283, 3157246719905273, 135938652662043977, 6275929675565965599, 309242148569525451140, 16197470691388774460758, 898619766673014862321176, 52639402023471657682257626

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 1 through a(3) = 8 graphs:
  {}  {}  {}       {}
          {{1,2}}  {{1,2}}
                   {{1,3}}
                   {{2,3}}
                   {{1,2},{1,3}}
                   {{1,2},{2,3}}
                   {{1,3},{2,3}}
                   {{1,2},{1,3},{2,3}}

The case of equality is A367862, covering case of A116508, also A367863.
The covering case is A369191, for loop-graphs A369194.
The version counting all vertices is A369192.
The version for loop-graphs is A369196, counting all vertices A066383.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable graphs, covering A367869.
A367867 counts non-choosable graphs, covering A367868.
Cf. A000169, A000272, A000666, A001187, A006649, A053530, A129271, A143543, A322661, A367916.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[#]<=Length[Union@@#]&]],{n,0,5}]

Binomial transform of A369191.

A368186 Number of n-covers of an unlabeled n-set.

Original entry on oeis.org

1, 1, 2, 9, 87, 1973, 118827, 20576251, 10810818595, 17821875542809, 94589477627232498, 1651805220868992729874, 96651473179540769701281003, 19238331716776641088273777321428, 13192673305726630096303157068241728202, 31503323006770789288222386469635474844616195

Offset: 0

Gus Wiseman, Dec 19 2023

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 set-systems:
  {{1}}  {{1},{2}}    {{1},{2},{3}}
         {{1},{1,2}}  {{1},{2},{1,3}}
                      {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{2},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1,2},{1,3},{2,3}}
                      {{1},{2,3},{1,2,3}}
                      {{1,2},{1,3},{1,2,3}}

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

The labeled version is A054780, ranks A367917, non-covering A367916.
The case of graphs is A006649, labeled A367863, cf. A116508, A367862.
The case of connected graphs is A001429, labeled A057500.
Covers with any number of edges are counted by A003465, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A002494, A006126, A055130, A133686, A140638, A305000, A317795, A326754, A367901, A367902, A367903.

brute[m_]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i, p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}];
Table[Length[Union[First[Sort[brute[#]]]& /@ Select[Subsets[Rest[Subsets[Range[n]]],{n}], Union@@#==Range[n]&]]], {n,0,3}]

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}
K(q, t)={2^sum(j=1, #q, gcd(t, q[j])) - 1}
G(n,m)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, m, K(q,t)*x^t/t, O(x*x^m))); s+=permcount(q)*exp(g - subst(g,x,x^2))); s/n!)}
a(n)=if(n ==0, 1, polcoef(G(n,n) - G(n-1,n), n)) \\ Andrew Howroyd, Jan 03 2024

a(n) = A055130(n, n) for n > 0. - Andrew Howroyd, Jan 03 2024

Terms a(6) and beyond from Andrew Howroyd, Jan 03 2024

A369144 Number of labeled simple graphs with n edges covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 90, 4935, 200970, 7636860, 291089610, 11459170800, 471932476290, 20447369179380, 933942958593645, 44981469288560805, 2282792616992648670, 121924195590795244920, 6843305987751060036720, 403003907531795513467260, 24861219342100679072572470

Offset: 0

Gus Wiseman, Jan 21 2024

nonn

			The term a(6) = 90 counts all permutations of the (non-connected) graph {{1,2},{1,3},{1,4},{2,3},{2,4},{5,6}}.

Andrew Howroyd, Table of n, a(n) for n = 0..200

The covering complement is counted by A137916.
Without the choice condition we have A367863, covering case of A116508.
Allowing any number of edges gives A367868, covering case of A367867.
With loops we have A368730, covering case of A368596, unlabeled A368835.
This is the covering case of A369143.
A003465 counts covering set-systems, unlabeled A055621.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A058891 counts set-systems, unlabeled A000612.
A322661 counts covering loop-graphs, connected A062740.
Cf. A000085, A057500, A129271, A133686, A333331, A367869, A367901, A367903, A367907, A368600.

Table[Length[Select[Subsets[Subsets[Range[n],{2}], {n}],Union@@#==Range[n]&&Length[Select[Tuples[#], UnsameQ@@#&]]==0&]],{n,0,6}]

a(n) = A367863(n) - A137916(n). - Andrew Howroyd, Feb 02 2024

a(8) onwards from Andrew Howroyd, Feb 02 2024

cp's OEIS Frontend

A368951 Number of connected labeled graphs with n edges and n vertices and with loops allowed.

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A368730 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, such that it is not possible to choose a different element from each.

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A368924 Triangle read by rows where T(n,k) is the number of labeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different vertex from each edge.

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A367917 BII-numbers of set-systems with the same number of edges as covered vertices.

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A369143 Number of labeled simple graphs with n edges and n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

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A369196 Number of labeled loop-graphs with n vertices and at most as many edges as covered vertices.

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A370169 Number of unlabeled loop-graphs covering n vertices with at most n edges.

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A369193 Number of labeled simple graphs with n vertices and at most as many edges as covered (non-isolated) vertices.

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