A116508 -id:A116508 - cp's OEIS frontend

A066383 a(n) = Sum_{k=0..n} C(n*(n+1)/2,k).

1, 2, 7, 42, 386, 4944, 82160, 1683218, 40999516, 1156626990, 37060382822, 1328700402564, 52676695500313, 2287415069586304, 107943308165833912, 5499354613856855290, 300788453960472434648, 17577197510240126035698, 1092833166733915284972350

Offset: 0

N. J. A. Sloane, Dec 23 2001

nonn

Number of labeled loop-graphs with n vertices and at most n edges. - Gus Wiseman, Feb 14 2024

			From _Gus Wiseman_, Feb 14 2024: (Start)
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}}
(End)

Harry J. Smith, Table of n, a(n) for n = 0..100

The case of equality is A014068, covering A368597.
The loopless version is A369192, covering A369191.
The covering case is A369194, minimal case A001862.
Counting only covered vertices gives A369196, without loops A369193.
The connected covering case is A369197, without loops A129271.
The unlabeled version is A370168, covering A370169.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000085, A000666, A005703, A062740, A116508, A367862, A367863, A367916, A368927, A369141, A369199.

f[n_] := Sum[Binomial[n (n + 1)/2, k], {k, 0, n}]; Array[f, 21, 0] (* Vincenzo Librandi, May 06 2016 *)
Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]],Length[#]<=n&]],{n,0,5}] (* Gus Wiseman, Feb 14 2024 *)

{ for (n=0, 100, a=0; for (k=0, n, a+=binomial(n*(n + 1)/2, k)); write("b066383.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 12 2010

from math import comb
def A066383(n): return sum(comb(comb(n+1,2),k) for k in range(n+1)) # Chai Wah Wu, Jul 10 2024

a(n) = 2^(n*(n+1)/2) - binomial(n*(n+1)/2,n+1)*2F1(1,(-n^2+n+2)/2;n+2;-1) = A006125(n) - A116508(n+1) * 2F1(1,(-n^2+n+2)2;n+2;-1), where 2F1(a,b;c;x) is the hypergeometric function. - Ilya Gutkovskiy, May 06 2016

a(n) ~ exp(n) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, Feb 20 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.

A001434 Number of graphs with n nodes and n edges.

Original entry on oeis.org

1, 0, 0, 1, 2, 6, 21, 65, 221, 771, 2769, 10250, 39243, 154658, 628635, 2632420, 11353457, 50411413, 230341716, 1082481189, 5228952960, 25945377057, 132140242356, 690238318754, 3694876952577, 20252697246580, 113578669178222, 651178533855913, 3813856010041981

Offset: 0

N. J. A. Sloane

The labeled version is A116508. - Gus Wiseman, Feb 22 2024

			From _Gus Wiseman_, Feb 22 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 6 graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}
                        {12,13,24,34}  {12,13,14,23,24}
                                       {12,13,14,23,25}
                                       {12,13,14,23,45}
                                       {12,13,14,25,35}
                                       {12,13,24,35,45}
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.
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).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 1..40 from Sean A. Irvine)
CombOS - Combinatorial Object Server, Generate graphs
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967.

The connected case is A001429, labeled A057500.
The covering case is A006649, labeled A367863.
Diagonal n = k of A008406.
The labeled version is A116508.
The version with loops is A368598, connected A368983.
Allowing up to n edges gives A370315, labeled A369192.
A000088 counts unlabeled graphs, labeled A006125.
A001349 counts unlabeled connected graphs, labeled A001187.
A002494 counts unlabeled covering graphs, labeled A006129.
Cf. A000055, A005703, A014068, A054924, A137916, A137917, A236570, A369201.

(* first do *) Needs["Combinatorica`"] (* then *) Table[ NumberOfGraphs[n, n], {n, 24}] (* Robert G. Wilson v, Mar 22 2011 *)
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 /@ Subsets[Subsets[Range[n],{2}],{n}]]],{n,0,5}] (* Gus Wiseman, Feb 22 2024 *)

a(n) = polcoef(G(n, O(x*x^n)), n) \\ G defined in A008406. - Andrew Howroyd, Feb 02 2024

More terms from Vladeta Jovovic, Jan 07 2000

a(0)=1 prepended by Andrew Howroyd, Feb 02 2024

A368984 Number of graphs with loops (symmetric relations) on n unlabeled vertices in which each connected component has an equal number of vertices and edges.

Original entry on oeis.org

1, 1, 2, 5, 12, 29, 75, 191, 504, 1339, 3610, 9800, 26881, 74118, 205706, 573514, 1606107, 4513830, 12727944, 35989960, 102026638, 289877828, 825273050, 2353794251, 6724468631, 19239746730, 55123700591, 158133959239, 454168562921, 1305796834570, 3758088009136

Offset: 0

Andrew Howroyd, Jan 11 2024

nonn

The graphs considered here can have loops but not parallel edges.

Also the number of unlabeled loop-graphs with n edges and n vertices such that it is possible to choose a different vertex from each edge. - Gus Wiseman, Jan 25 2024

			Representatives of the a(3) = 5 graphs are:
   {{1,2}, {1,3}, {2,3}},
   {{1}, {1,2}, {1,3}},
   {{1}, {1,2}, {2,3}},
   {{1}, {2}, {2,3}},
   {{1}, {2}, {3}}.
The graph with 4 vertices and edges {{1}, {2}, {1,2}, {3,4}} is included by A368599 but not by this sequence.

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

The case of a unique choice is A000081.
Without loops we have A137917, labeled A137916.
The labeled version appears to be A333331.
Without the choice condition we have A368598, covering A368599.
The complement is counted by A368835, labeled A368596 (covering A368730).
Row sums of A368926, labeled A368924.
The connected case is A368983.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, connected A001187, unlabeled A002494.
A322661 counts labeled covering loop-graphs, connected A062740.
Cf. A014068, A057500, A116508, A129271, A133686, A367863, A367869, A367902, A368597, A368601, A368836.

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}],Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

Euler transform of A368983.

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

A368835 Number of unlabeled n-edge loop-graphs with at most n vertices such that it is not possible to choose a different vertex from each edge.

Original entry on oeis.org

0, 0, 0, 1, 5, 23, 98, 394, 1560, 6181, 24655, 99701, 410513, 1725725, 7423757, 32729320, 148027044, 687188969, 3275077017, 16022239940, 80431483586, 414094461610, 2185052929580, 11808696690600, 65312048149993, 369408792148714, 2135111662435080, 12601466371445619

Offset: 0

Gus Wiseman, Jan 13 2024

nonn

			Non-isomorphic representatives of the a(4) = 5 loop-graphs:
  {{1,1},{2,2},{3,3},{1,2}}
  {{1,1},{2,2},{1,2},{1,3}}
  {{1,1},{2,2},{1,2},{3,4}}
  {{1,1},{2,2},{1,3},{2,3}}
  {{1,1},{1,2},{1,3},{2,3}}

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

The case of a unique choice is A000081, row sums of A106234.
The labeled version is A368596, covering A368730.
Without the choice condition we have A368598.
The complement is A368984, row sums of A368926.
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 (without singletons A016031), unlabeled A000612.
A322661 counts labeled covering half-loop-graphs, connected A062740.
Cf. A116508, A129271, A133686, A137916, A137917, A333331, A367863, A368836, A368924, A368927.

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

a(n) = A368598(n) - A368984(n). - Andrew Howroyd, Jan 14 2024

a(8) onwards from Andrew Howroyd, Jan 14 2024

A369140 Number of labeled loop-graphs covering {1..n} such that it is possible to choose a different vertex from each edge (choosable).

Original entry on oeis.org

1, 1, 4, 23, 193, 2133, 29410, 486602, 9395315, 207341153, 5147194204, 141939786588, 4304047703755, 142317774817901, 5095781837539766, 196403997108015332, 8106948166404074281, 356781439557643998591, 16675999433772328981216, 824952192369049982670686

Offset: 0

Gus Wiseman, Jan 20 2024

nonn

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

			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}}

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

For a unique choice we have A000272, covering case of A088957.
Without the choice condition we have A322661, unlabeled A322700.
For exactly n edges we have A333331 (maybe), complement A368596.
The case without loops is A367869, covering case of A133686.
This is the covering case of A368927.
The complement is counted by A369142, covering case of A369141.
The unlabeled version is the first differences of A369145.
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.
A367862 counts graphs with n vertices and n edges, covering A367863.
Cf. A000169, A000666, A003465, A006649, A062740, A116508, A137916, A368924, A368984, A369194.

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

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

Inverse binomial transform of A368927.
Exponential transform of A369197.
E.g.f.: exp(-x)*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(6) onwards from Andrew Howroyd, Feb 02 2024

A369191 Number of labeled simple graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 0, 1, 4, 34, 387, 5686, 102084, 2162168, 52693975, 1450876804, 44509105965, 1504709144203, 55563209785167, 2224667253972242, 95984473918245388, 4439157388017620554, 219067678811211857307, 11489425098298623161164, 638159082104453330569185

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

Row-sums of left portion of A054548.

			The a(0) = 1 through a(3) = 4 graphs:
  {}  .  {{1,2}}  {{1,2},{1,3}}
                  {{1,2},{2,3}}
                  {{1,3},{2,3}}
                  {{1,2},{1,3},{2,3}}

The minimal case is A053530.
The connected case is A129271, unlabeled version A005703.
The case of equality is A367863, covering case of A367862.
This is the covering case of A369192, or A369193 for covered vertices.
The version for loop-graphs is A369194.
The unlabeled version is A370316.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A057500 counts connected graphs with n vertices and n edges.
A133686 counts choosable graphs, covering A367869.
A367867 counts non-choosable graphs, covering A367868.
Cf. A000169, A000272, A000666, A001429, A003465, A006649, A143543, A116508, A140638, A322661.

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

Inverse binomial transform of A369193.

A369192 Number of labeled simple graphs with n vertices and at most n edges (not necessarily covering).

Original entry on oeis.org

1, 1, 2, 8, 57, 638, 9949, 198440, 4791323, 135142796, 4346814276, 156713948672, 6251579884084, 273172369790743, 12969420360339724, 664551587744173992, 36543412829258260135, 2146170890448154922648, 134053014635659737513358, 8872652968135849629240560

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 version for loop-graphs is A066383, covering A369194.
The case of equality is A116508, covering A367863, also A367862.
The connected case is A129271, unlabeled A005703.
The covering case is A369191, minimal case A053530.
Counting only covered vertices gives A369193.
A006125 counts 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, A057500, A143543.

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

from math import comb
def A369192(n): return sum(comb(comb(n,2),k) for k in range(n+1)) # Chai Wah Wu, Jul 14 2024

a(n) = Sum_{k=0..n} binomial(binomial(n,2),k).

A054780 Number of n-covers of a labeled n-set.

Original entry on oeis.org

1, 1, 3, 32, 1225, 155106, 63602770, 85538516963, 386246934638991, 6001601072676524540, 327951891446717800997416, 64149416776011080449232990868, 45546527789182522411309599498741023, 118653450898277491435912500458608964207578

Offset: 0

Vladeta Jovovic, May 21 2000

Also, number of n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct, up to permutation of rows.

			From _Gus Wiseman_, Dec 19 2023: (Start)
Number of ways to choose n nonempty sets with union {1..n}. For example, the a(3) = 32 covers are:
  {1}{2}{3}  {1}{2}{13}  {1}{2}{123}  {1}{12}{123}  {12}{13}{123}
             {1}{2}{23}  {1}{3}{123}  {1}{13}{123}  {12}{23}{123}
             {1}{3}{12}  {1}{12}{13}  {1}{23}{123}  {13}{23}{123}
             {1}{3}{23}  {1}{12}{23}  {2}{12}{123}
             {2}{3}{12}  {1}{13}{23}  {2}{13}{123}
             {2}{3}{13}  {2}{3}{123}  {2}{23}{123}
                         {2}{12}{13}  {3}{12}{123}
                         {2}{12}{23}  {3}{13}{123}
                         {2}{13}{23}  {3}{23}{123}
                         {3}{12}{13}  {12}{13}{23}
                         {3}{12}{23}
                         {3}{13}{23}
(End)

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

Main diagonal of A055154.
Cf. A048291, A088310, A181230, A259763.
Covers with any number of edges are counted by A003465, unlabeled A055621.
Connected graphs of this type are counted by A057500, unlabeled A001429.
This is the covering case of A136556.
The case of graphs is A367863, covering case of A116508, unlabeled A006649.
Binomial transform is A367916.
These set-systems have ranks A367917.
The unlabeled version is A368186.
A006129 counts covering graphs, connected A001187, unlabeled A002494.
A046165 counts minimal covers, ranks A309326.
Cf. A006126, A058891, A059201, A133686, A323816, A323817, A323818, A326754, A367862, A367901.

Join[{1}, Table[Sum[StirlingS1[n+1, k+1]*(2^k - 1)^n, {k, 0, n}]/n!, {n, 1, 15}]] (* Vaclav Kotesovec, Jun 04 2022 *)
Table[Length[Select[Subsets[Rest[Subsets[Range[n]]],{n}],Union@@#==Range[n]&]],{n,0,4}] (* Gus Wiseman, Dec 19 2023 *)

a(n) = sum(k=0, n, (-1)^k*binomial(n, k)*binomial(2^(n-k)-1, n)) \\ Andrew Howroyd, Jan 20 2024

a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*binomial(2^(n-k)-1, n).
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n+1, k+1)*(2^k-1)^n.
G.f.: Sum_{n>=0} log(1+(2^n-1)*x)^n/((1+(2^n-1)*x)*n!). - Paul D. Hanna and Vladeta Jovovic, Jan 16 2008
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jun 04 2022
Inverse binomial transform of A367916. - Gus Wiseman, Dec 19 2023

cp's OEIS Frontend

A066383 a(n) = Sum_{k=0..n} C(n*(n+1)/2,k).

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

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A001434 Number of graphs with n nodes and n edges.

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A368984 Number of graphs with loops (symmetric relations) on n unlabeled vertices in which each connected component has an equal number of vertices and 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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A368835 Number of unlabeled n-edge loop-graphs with at most n vertices such that it is not possible to choose a different vertex from each edge.

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A369140 Number of labeled loop-graphs covering {1..n} such that it is possible to choose a different vertex from each edge (choosable).

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