A369140 -id:A369140 - cp's OEIS frontend

A369199 Irregular triangle read by rows where T(n,k) is the number of labeled loop-graphs covering n vertices with k edges.

1, 0, 1, 0, 1, 3, 1, 0, 0, 6, 17, 15, 6, 1, 0, 0, 3, 46, 150, 228, 206, 120, 45, 10, 1, 0, 0, 0, 45, 465, 1803, 3965, 5835, 6210, 4955, 2998, 1365, 455, 105, 15, 1, 0, 0, 0, 15, 645, 5991, 27364, 79470, 165555, 264050, 334713, 344526, 291200, 202860, 116190, 54258, 20349, 5985, 1330, 210, 21, 1

Offset: 0

Gus Wiseman, Jan 18 2024

			Triangle begins:
   1
   0   1
   0   1   3   1
   0   0   6  17  15   6   1
   0   0   3  46 150 228 206 120  45  10   1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
  {1,23}   {1,2,3}     {1,2,3,12}    {1,2,3,12,13}   {1,2,3,12,13,23}
  {2,13}   {1,2,13}    {1,2,3,13}    {1,2,3,12,23}
  {3,12}   {1,2,23}    {1,2,3,23}    {1,2,3,13,23}
  {12,13}  {1,3,12}    {1,2,12,13}   {1,2,12,13,23}
  {12,23}  {1,3,23}    {1,2,12,23}   {1,3,12,13,23}
  {13,23}  {1,12,13}   {1,2,13,23}   {2,3,12,13,23}
           {1,12,23}   {1,3,12,13}
           {1,13,23}   {1,3,12,23}
           {2,3,12}    {1,3,13,23}
           {2,3,13}    {1,12,13,23}
           {2,12,13}   {2,3,12,13}
           {2,12,23}   {2,3,12,23}
           {2,13,23}   {2,3,13,23}
           {3,12,13}   {2,12,13,23}
           {3,12,23}   {3,12,13,23}
           {3,13,23}
           {12,13,23}

Andrew Howroyd, Table of n, a(n) for n = 0..1560 (rows 0..20)

The version without loops is A054548.
This is the covering case of A084546.
Column sums are A173219.
Row sums are A322661, unlabeled A322700.
The connected case is A369195, without loops A062734.
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.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000666, A006649, A062740, A066383, A133686, A368597, A369196, A369197.

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

T(n)={[Vecrev(p) | p<-Vec(serlaplace(exp(-x + O(x*x^n))*(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!)))) ]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 02 2024

E.g.f.: exp(-x) * (Sum_{j >= 0} (1 + y)^binomial(j+1, 2)*x^j/j!). - Andrew Howroyd, Feb 02 2024

A088957 Hyperbinomial transform of the sequence of 1's.

Original entry on oeis.org

1, 2, 6, 29, 212, 2117, 26830, 412015, 7433032, 154076201, 3608522954, 94238893883, 2715385121740, 85574061070045, 2928110179818478, 108110945014584623, 4284188833355367440, 181370804507130015569, 8169524599872649117330, 390114757072969964280163

Offset: 0

Paul D. Hanna, Oct 26 2003

nonn

See A088956 for the definition of the hyperbinomial transform.
a(n) is the number of partial functions on {1,2,...,n} that are endofunctions with no cycles of length > 1.  The triangle A088956 classifies these functions according to the number of undefined elements in the domain.  The triangle A144289 classifies these functions according to the number of edges in their digraph representation (considering the empty function to have 1 edge).  The triangle A203092 classifies these functions according to the number of connected components. - Geoffrey Critzer, Dec 29 2011
a(n) is the number of rooted subtrees (for a fixed root) in the complete graph on n+1 vertices: a(3) = 29 is the number of rooted subtrees in K_4: 1 of size 1, 3 of size 2, 9 of size 3, and 16 spanning subtrees. - Alex Chin, Jul 25 2013 [corrected by Marko Riedel, Mar 31 2019]
From Gus Wiseman, Jan 28 2024: (Start)
Also the number of labeled loop-graphs on n vertices such that it is possible to choose a different vertex from each edge in exactly one way. For example, the a(3) = 29 uniquely choosable loop-graphs (loops shown as singletons) are:
  {}  {1}  {1,2}  {1,12}   {1,2,13}  {1,12,13}
      {2}  {1,3}  {1,13}   {1,2,23}  {1,12,23}
      {3}  {2,3}  {2,12}   {1,3,12}  {1,13,23}
                  {2,23}   {1,3,23}  {2,12,13}
                  {3,13}   {2,3,12}  {2,12,23}
                  {3,23}   {2,3,13}  {2,13,23}
                  {1,2,3}            {3,12,13}
                                     {3,12,23}
                                     {3,13,23}
(End)

			a(5) = 2117 = 1296 + 625 + 160 + 30 + 5 + 1 = sum of row 5 of triangle A088956.

Alois P. Heinz, Table of n, a(n) for n = 0..387
Marko Riedel et al., Proof of e.g.f. of sequence.

Cf. A088956 (triangle).
Row sums of A144289. - Alois P. Heinz, Jun 01 2009
Cf. A086331, A000169.
Column k=1 of A144303. - Alois P. Heinz, Oct 30 2012
The covering case is A000272, also the case of exactly n edges.
Without the choice condition we have A006125 (shifted left).
The unlabeled version is A087803.
The choosable version is A368927, covering A369140, loopless A133686.
The non-choosable version is A369141, covering A369142, loopless A367867.
Cf. A000081, A000085, A057500, A062740, A137916, A277473, A322661, A367904, A368596, A368597, A368924.

a088957 = sum . a088956_row  -- Reinhard Zumkeller, Jul 07 2013

a:= n-> add((n-j+1)^(n-j-1)*binomial(n,j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 30 2012

nn = 16; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[Exp[x] Exp[t], {x, 0, nn}], x]  (* Geoffrey Critzer, Dec 29 2011 *)
With[{nmax = 50}, CoefficientList[Series[-LambertW[-x]*Exp[x]/x, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 14 2017 *)

x='x+O('x^10); Vec(serlaplace(-lambertw(-x)*exp(x)/x)) \\ G. C. Greubel, Nov 14 2017

a(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k).
E.g.f.: A(x) = exp(x+sum(n>=1, n^(n-1)*x^n/n!)).
E.g.f.: -LambertW(-x)*exp(x)/x. - Vladeta Jovovic, Oct 27 2003
a(n) ~ exp(1+exp(-1))*n^(n-1). - Vaclav Kotesovec, Jul 08 2013
Binomial transform of A000272. - Gus Wiseman, Jan 25 2024

A368927 Number of labeled loop-graphs covering a subset of {1..n} such that it is possible to choose a different vertex from each edge.

Original entry on oeis.org

1, 2, 7, 39, 314, 3374, 45630, 744917, 14245978, 312182262, 7708544246, 211688132465, 6397720048692, 210975024924386, 7537162523676076, 289952739051570639, 11949100971787370300, 525142845422124145682, 24515591201199758681892, 1211486045654016217202663

Offset: 0

Gus Wiseman, Jan 15 2024

nonn

These are loop-graphs where every connected component has a number of edges less than or equal to the number of vertices. Also loop-graphs with at most one cycle (unicyclic) in each connected component.

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

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

Without the choice condition we have A006125.
The case of a unique choice is A088957, unlabeled A087803.
The case without loops is A133686, complement A367867, covering A367869.
For exactly n edges and no loops we have A137916, unlabeled A137917.
For exactly n edges we have A333331 (maybe), complement A368596.
For edges of any positive size we have A367902, complement A367903.
The covering case is A369140, complement A369142.
The complement is counted by A369141.
The complement for n edges and no loops is A369143, covering A369144.
The unlabeled version is A369145, complement A369146.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts labeled covering loop-graphs, connected A062740.
Cf. A000169, A000272, A000666, A014068, A057500, A116508, A367863, A368601, A368924, A368984, A369200.

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

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(exp(3*t/2 - 3*t^2/4)/sqrt(1-t) ))} \\ Andrew Howroyd, Feb 02 2024

Binomial transform of A369140.
Exponential transform of A369197 with A369197(1) = 2.
E.g.f.: exp(3*T(x)/2 - 3*T(x)^2/4)/sqrt(1-T(x)), where T(x) is the e.g.f. of A000169. - Andrew Howroyd, Feb 02 2024

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

A369141 Number of labeled loop-graphs covering a subset of {1..n} such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 1, 25, 710, 29394, 2051522, 267690539, 68705230758, 35184059906570, 36028789310419722, 73786976083150073999, 302231454897259573627852, 2475880078570549574773324062, 40564819207303333310731978895956, 1329227995784915872613854321228773937

Offset: 0

Gus Wiseman, Jan 20 2024

nonn

Also labeled loop-graphs having at least one connected component containing more edges than vertices.

			The a(0) = 0 through a(3) = 25 loop-graphs (loops shown as singletons):
  .  .  {{1},{2},{1,2}}  {{1},{2},{1,2}}
                         {{1},{3},{1,3}}
                         {{2},{3},{2,3}}
                         {{1},{2},{3},{1,2}}
                         {{1},{2},{3},{1,3}}
                         {{1},{2},{3},{2,3}}
                         {{1},{2},{1,2},{1,3}}
                         {{1},{2},{1,2},{2,3}}
                         {{1},{2},{1,3},{2,3}}
                         {{1},{3},{1,2},{1,3}}
                         {{1},{3},{1,2},{2,3}}
                         {{1},{3},{1,3},{2,3}}
                         {{2},{3},{1,2},{1,3}}
                         {{2},{3},{1,2},{2,3}}
                         {{2},{3},{1,3},{2,3}}
                         {{1},{1,2},{1,3},{2,3}}
                         {{2},{1,2},{1,3},{2,3}}
                         {{3},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3}}
                         {{1},{2},{3},{1,2},{2,3}}
                         {{1},{2},{3},{1,3},{2,3}}
                         {{1},{2},{1,2},{1,3},{2,3}}
                         {{1},{3},{1,2},{1,3},{2,3}}
                         {{2},{3},{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

Without the choice condition we have A006125, unlabeled A000088.
The case of a unique choice is A088957, unlabeled A087803.
The case without loops is A367867, covering A367868.
For edges of any positive size we have A367903, complement A367902.
For exactly n edges we have A368596, complement A333331 (maybe).
The complement is counted by A368927, covering A369140.
The covering case is A369142.
For n edges and no loops we have A369143, covering A369144.
The unlabeled version is A369146 (covering A369147), complement A369145.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable graphs, covering A367869.
A322661 counts labeled covering loop-graphs, unlabeled A322700.
Cf. A000272, A000666, A006649, A062740, A116508, A137916, A368924, A368984, A369194.

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

Binomial transform of A369142.

a(n) = A006125(n + 1) - A368927(n). - Andrew Howroyd, Feb 02 2024

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

A369197 Number of labeled connected loop-graphs with n vertices, none isolated, and at most n edges.

Original entry on oeis.org

1, 1, 3, 13, 95, 972, 12732, 202751, 3795864, 81609030, 1980107840, 53497226337, 1592294308992, 51758060711792, 1824081614046720, 69272000503031475, 2819906639193992192, 122488526636380368714, 5654657850859704139776, 276462849597009068108405, 14270030377126199463936000

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

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

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

The minimal case is A000272.
Connected case of A066383 and A369196, loopless A369192 and A369193.
The loopless case is A129271, connected case of A369191.
The case of equality is A368951, connected case of A368597.
This is the connected case of A369194.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts (simple) graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A062740 counts connected loop-graphs.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000666, A001429, A005703, A006649, A057500, A116508, A140638, A143543, A367863, A369145.

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + 3*t/2 - 3*t^2/4 + 1 - x))} \\ Andrew Howroyd, Feb 02 2024

Logarithmic transform of A368927.
From Andrew Howroyd, Feb 02 2024: (Start)
a(n) = A000169(n) + A129271(n).
E.g.f.: log(1/(1-T(x)))/2 + 3*T(x)/2 - 3*T(x)^2/4 + 1 - x, where T(x) is the e.g.f. of A000169. (End)

a(0) changed to 1 and a(7) onwards from Andrew Howroyd, Feb 02 2024

A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 1, 4, 23, 199, 2313, 34015, 606407, 12712643, 306407645, 8346154699, 253476928293, 8490863621050, 310937199521774, 12356288017546937, 529516578044589407, 24339848939829286381, 1194495870124420574751, 62332449791125883072149, 3446265450868329833016605

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

Row-sums of left portion of A369199.

			The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{2},{1,3}}
             {{1},{1,2}}  {{3},{1,2}}
             {{2},{1,2}}  {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1},{2},{1,3}}
                          {{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}}

The minimal case is A001862, without loops A053530.
This is the covering case of A066383 and A369196, cf. A369192 and A369193.
The case of equality is A368597, without loops A367863.
The version without loops is A369191.
The connected case is A369197, without loops A129271.
The unlabeled version is A370169, equality A368599, non-covering A368598.
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.
A133686 counts choosable graphs, covering A367869.
A322661 counts covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable graphs, covering A367868.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, A000666, A003465, A005703, A006649, A057500, A062740, A116508, A367862, A367916.

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

Inverse binomial transform of A369196.

A369142 Number of labeled loop-graphs covering {1..n} such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 1, 22, 616, 26084, 1885323, 253923163, 66619551326, 34575180977552, 35680008747431929, 73392583275070667841, 301348381377662031986734, 2471956814761854578316988092, 40530184362443276558060719358471, 1328619783326799871747200601484790193

Offset: 0

Gus Wiseman, Jan 20 2024

nonn

Also labeled loop-graphs covering n vertices with at least one connected component containing more edges than vertices.

			The a(0) = 0 through a(3) = 22 loop-graphs (loops shown as singletons):
  .  .  {{1},{2},{1,2}}  {{1},{2},{3},{1,2}}
                         {{1},{2},{3},{1,3}}
                         {{1},{2},{3},{2,3}}
                         {{1},{2},{1,2},{1,3}}
                         {{1},{2},{1,2},{2,3}}
                         {{1},{2},{1,3},{2,3}}
                         {{1},{3},{1,2},{1,3}}
                         {{1},{3},{1,2},{2,3}}
                         {{1},{3},{1,3},{2,3}}
                         {{2},{3},{1,2},{1,3}}
                         {{2},{3},{1,2},{2,3}}
                         {{2},{3},{1,3},{2,3}}
                         {{1},{1,2},{1,3},{2,3}}
                         {{2},{1,2},{1,3},{2,3}}
                         {{3},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3}}
                         {{1},{2},{3},{1,2},{2,3}}
                         {{1},{2},{3},{1,3},{2,3}}
                         {{1},{2},{1,2},{1,3},{2,3}}
                         {{1},{3},{1,2},{1,3},{2,3}}
                         {{2},{3},{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

The version for a unique choice is A000272, unlabeled A000055.
Without the choice condition we have A006125, unlabeled A000088.
The case without loops is A367868, covering case of A367867.
For exactly n edges we have A368730, covering case of A368596.
The complement is counted by A369140, covering case of A368927.
This is the covering case of A369141.
For n edges and no loops we have A369144, covering A369143.
The unlabeled version is A369147, covering case of A369146.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable graphs, unlabeled A005703.
A133686 counts choosable graphs, covering A367869.
A322661 counts covering loop-graphs, connected A062740, unlabeled A322700.
A367902 counts choosable set-systems, complement A367903.
Cf. A000666, A003465, A006649, A116508, A137916, A333331, A368600, A368924, A368984, A369145, A369194.

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

Inverse binomial transform of A369141.

a(n) = A322661(n) - A369140(n). - Andrew Howroyd, Feb 02 2024

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

A369146 Number of unlabeled loop-graphs with up to 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, 1, 8, 60, 471, 4911, 78797, 2207405, 113740613, 10926218807, 1956363413115, 652335084532025, 405402273420833338, 470568642161119515627, 1023063423471189429817807, 4178849203082023236054797465, 32168008290073542372004072630072, 468053896898117580623237189882068990

Offset: 0

Gus Wiseman, Jan 22 2024

nonn

			The a(0) = 0 through a(3) = 8 loop-graphs (loops shown as singletons):
  .  .  {{1},{2},{1,2}}  {{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

Without the choice condition we have A000666, labeled A006125 (shifted).
For a unique choice we have A087803, labeled A088957.
The case without loops is A140637, labeled A367867 (covering A367868).
For exactly n edges we have A368835, labeled A368596.
The labeled complement is A368927, covering A369140.
The labeled version is A369141, covering A369142.
The complement is counted by A369145, covering A369200.
The covering case is A369147.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A129271 counts connected choosable simple graphs, unlabeled A005703.
A322661 counts labeled covering loop-graphs, unlabeled A322700.
Cf. A000088, A000612, A006649, A062740, A134964, A137917, A140638, A368600, A368984, 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}]], Select[Tuples[#],UnsameQ@@#&]=={}&]]],{n,0,4}]

Partial sums of A369147.

a(n) = A000666(n) - A369145(n). - Andrew Howroyd, Feb 02 2024

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

A369145 Number of unlabeled loop-graphs with up to n vertices such that it is possible to choose a different vertex from each edge (choosable).

Original entry on oeis.org

1, 2, 5, 12, 30, 73, 185, 467, 1207, 3147, 8329, 22245, 60071, 163462, 448277, 1236913, 3432327, 9569352, 26792706, 75288346, 212249873, 600069431, 1700826842, 4831722294, 13754016792, 39224295915, 112048279650, 320563736148, 918388655873, 2634460759783, 7566000947867

Offset: 0

Gus Wiseman, Jan 22 2024

nonn

a(n) is the number of graphs with loops on n unlabeled vertices with every connected component having no more edges than vertices. - Andrew Howroyd, Feb 02 2024

			The a(0) = 1 through a(3) = 12 loop-graphs (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},{1,3}}
                          {{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..500

Without the choice condition we get A000666, labeled A006125 (shifted left).
The case of a unique choice is A087803, labeled A088957.
Without loops we have A134964, labeled A133686 (covering A367869).
For exactly n edges and no loops we have A137917, labeled A137916.
The labeled version is A368927, covering A369140.
The labeled complement is A369141, covering A369142.
For exactly n edges we have A368984, labeled A333331 (maybe).
The complement for exactly n edges is A368835, labeled A368596.
The complement is counted by A369146, labeled A369141 (covering A369142).
The covering case is A369200.
The complement for exactly n edges and no loops is A369201, labeled A369143.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
A322661 counts labeled covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable labeled graphs, covering A367868.
A368927 counts choosable labeled loop-graphs, covering A369140.
Cf. A000088, A006649, A062740, A140638, A368924, A369194.

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[Select[Tuples[#], UnsameQ@@#&]]!=0&]]],{n,0,4}]

Partial sums of A369200.

Euler transform of A369289. - Andrew Howroyd, Feb 02 2024

a(7) 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.

cp's OEIS Frontend

A369199 Irregular triangle read by rows where T(n,k) is the number of labeled loop-graphs covering n vertices with k edges.

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A088957 Hyperbinomial transform of the sequence of 1's.

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A368927 Number of labeled loop-graphs covering a subset of {1..n} such that it is possible to choose a different vertex from each edge.

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A369141 Number of labeled loop-graphs covering a subset of {1..n} such that it is not possible to choose a different vertex from each edge (non-choosable).

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A369197 Number of labeled connected loop-graphs with n vertices, none isolated, and at most n edges.

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A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

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A369142 Number of labeled loop-graphs covering {1..n} such that it is not possible to choose a different vertex from each edge (non-choosable).

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