A369191 -id:A369191 - cp's OEIS frontend

A057500 Number of connected labeled graphs with n edges and n nodes.

0, 0, 1, 15, 222, 3660, 68295, 1436568, 33779340, 880107840, 25201854045, 787368574080, 26667815195274, 973672928417280, 38132879409281475, 1594927540549217280, 70964911709203684440, 3347306760024413356032, 166855112441313024389625, 8765006377126199463936000

Offset: 1

Qing-Hu Hou and David C. Torney (dct(AT)lanl.gov), Sep 01 2000

Equivalently, number of connected unicyclic (i.e., containing one cycle) graphs on n labeled nodes. - Vladeta Jovovic, Oct 26 2004

a(n) is the number of trees on vertex set [n] = {1,2,...,n} rooted at 1 with one marked inversion (an inversion is a pair (i,j) with i > j and j a descendant of i in the tree). Here is a bijection from the title graphs (on [n]) to these marked trees. A title graph has exactly one cycle. There is a unique path from vertex 1 to this cycle, first meeting it at k, say (k may equal 1). Let i and j be the two neighbors of k in the cycle, with i the larger of the two. Delete the edge k<->j thereby forming a tree (in which j is a descendant of i) and take (i,j) as the marked inversion. To reverse this map, create a cycle by joining the smaller element of the marked inversion to the parent of the larger element. a(n) = binomial(n-1,2)*A129137(n). This is because, on the above marked trees, the marked inversion is uniformly distributed over 2-element subsets of {2,3,...,n} and so a(n)/binomial(n-1,2) is the number of trees on [n] (rooted at 1) for which (3,2) is an inversion. - David Callan, Mar 30 2007

			E.g., a(4)=15 because there are three different (labeled) 4-cycles and 12 different labeled graphs with a 3-cycle and an attached, external vertex.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.
C. L. Mallows, Letter to N. J. A. Sloane, 1980.
R. J. Riddell, Contributions to the theory of condensation, Dissertation, Univ. of Michigan, Ann Arbor, 1951.

Alois P. Heinz, Table of n, a(n) for n = 1..300 (terms n = 1..50 from Washington G. Bomfim)
Federico Ardila, Matthias Beck, Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 133.
S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, arXiv:math/9310236 [math.PR], 1993.
S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.
Younng-Jin Kim and Woong Kook, Winding number and Cutting number of Harmonic cycle, arXiv:1812.04930 [math.CO], 2018.
C. L. Mallows, Letter to N. J. A. Sloane, 1980
Marko Riedel et al., Non-isomorphic, connected, unicyclic graphs, Math Stackexchange, November 2018. (Proof of closed form by Cauchy Coefficient Formula / Lagrange Inversion.)
Chris Ying, Enumerating Unique Computational Graphs via an Iterative Graph Invariant, arXiv:1902.06192 [cs.DM], 2019.

A diagonal of A343088.
Cf. A000272 = labeled trees on n nodes; connected labeled graphs with n nodes and n+k edges for k=0..8: this sequence, A061540, A061541, A061542, A061543, A096117, A061544, A096150, A096224.
Cf. A001429 (unlabeled case), A052121.
For any number of edges we have A001187, unlabeled A001349.
This is the connected and covering case of A116508.
For #edges <= #nodes we have A129271, covering A367869.
For #edges > #nodes we have A140638, covering A367868.
This is the connected case of A367862 and A367863, unlabeled A006649.
The version with loops is A368951, unlabeled A368983.
This is the covering case of A370317.
Counting only covering vertices gives A370318.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
Cf. A062734, A098909, A123527, A129137, A133686, A143543, A323818, A367916, A367917, A369191, A369197.

egf:= -1/2*ln(1+LambertW(-x)) +1/2*LambertW(-x) -1/4*LambertW(-x)^2:
a:= n-> n!*coeff(series(egf, x, n+3), x, n):
seq(a(n), n=1..25);  # Alois P. Heinz, Mar 27 2013

nn=20; t=Sum[n^(n-1) x^n/n!, {n,1,nn}]; Drop[Range[0,nn]! CoefficientList[Series[Log[1/(1-t)]/2-t^2/4-t/2, {x,0,nn}], x], 1]  (* Geoffrey Critzer, Oct 07 2012 *)
a[n_] := (n-1)!*n^n/2*Sum[1/(n^k*(n-k)!), {k, 3, n}]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Jan 15 2014, after Vladeta Jovovic *)
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[#]==n&&Length[csm[#]]<=1&]],{n,0,5}] (* Gus Wiseman, Feb 19 2024 *)

# Warning: Floating point calculation. Adjust precision as needed!
from mpmath import mp, chop, gammainc
mp.dps = 200; mp.pretty = True
for n in (1..100):
    print(chop((n^(n-2)*(1-3*n)+exp(n)*gammainc(n+1, n)/n)/2))
# Peter Luschny, Jan 27 2016

The number of labeled connected graphs with n nodes and m edges is Sum_{k=1..n} (-1)^(k+1)/k*Sum_{n_1+n_2+..n_k=n, n_i>0} n!/(Product_{i=1..k} (n_i)!)* binomial(s, m), s=Sum_{i..k} binomial(n_i, 2). - Vladeta Jovovic, Apr 10 2001
E.g.f.: (1/2) Sum_{k>=3} T(x)^k/k, with T(x) = Sum_{n>=1} n^(n-1)/n! x^n. R. J. Riddell's thesis contains a closed-form expression for the number of connected graphs with m nodes and n edges. The present series applies to the special case m=n.
E.g.f.: -1/2*log(1+LambertW(-x))+1/2*LambertW(-x)-1/4*LambertW(-x)^2. - Vladeta Jovovic, Jul 09 2001
Asymptotic expansion (with xi=sqrt(2*Pi)): n^(n-1/2)*[xi/4-7/6*n^(-1/2)+xi/48* n^(-1)+131/270*n^(-3/2)+xi/1152*n^(-2)+4/2835*n^(-5/2)+O(n^(-3))]. - Keith Briggs, Aug 16 2004
Row sums of A098909: a(n) = (n-1)!*n^n/2*Sum_{k=3..n} 1/(n^k*(n-k)!). - Vladeta Jovovic, Oct 26 2004
a(n) = Sum_{k=0..C(n-1,2)} k*A052121(n,k). - Alois P. Heinz, Nov 29 2015
a(n) = (n^(n-2)*(1-3*n)+exp(n)*Gamma(n+1,n)/n)/2. - Peter Luschny, Jan 27 2016
a(n) = A062734(n,n+1) = A123527(n,n). - Gus Wiseman, Feb 19 2024

More terms from Vladeta Jovovic, Jul 09 2001

A129271 Number of labeled n-node connected graphs with at most one cycle.

Original entry on oeis.org

1, 1, 1, 4, 31, 347, 4956, 85102, 1698712, 38562309, 980107840, 27559801736, 849285938304, 28459975589311, 1030366840792576, 40079074477640850, 1666985134587145216, 73827334760713500233, 3468746291121007607808, 172335499299097826575564, 9027150377126199463936000

Offset: 0

Washington Bomfim, May 10 2008

The majority of those graphs of order 4 are trees since we have 16 trees and only 9 unicycles. See example.

Also connected graphs covering n vertices with at most n edges. The unlabeled version is A005703. - Gus Wiseman, Feb 19 2024

			a(4) = 16 + 3*3 = 31.
From _Gus Wiseman_, Feb 19 2024: (Start)
The a(0) = 1 through a(3) = 4 graph edge sets:
  {}  .  {{1,2}}  {{1,2},{1,3}}
                  {{1,2},{2,3}}
                  {{1,3},{2,3}}
                  {{1,2},{1,3},{2,3}}
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Dover, 2002, p. 2.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Wikipedia, PseudoForest.

Cf. A129137, A000272.
For any number of edges we have A001187, unlabeled A001349.
The unlabeled version is A005703.
The case of equality is A057500, covering A370317, cf. A370318.
The non-connected non-covering version is A133686.
The connected complement is A140638, unlabeled A140636, covering A367868.
The non-connected covering version is A367869 or A369191.
The version with loops is A369197, non-connected A369194.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A062734 counts connected graphs by number of edges.
Cf. A006649, A116508, A134964, A143543, A323818, A367862, A367863, A367867, A367916, A367917, A368951.

a := n -> `if`(n=0,1,((n-1)*exp(n)*GAMMA(n-1,n)+n^(n-2)*(3-n))/2):
seq(simplify(a(n)),n=0..16); # Peter Luschny, Jan 18 2016

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[ Log[1/(1-t)]/2+t/2-3t^2/4+1,{x,0,nn}],x]  (* Geoffrey Critzer, Mar 23 2013 *)

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + t/2 - 3*t^2/4 + 1))} \\ Andrew Howroyd, Nov 07 2019

a(0) = 1, for n >=1, a(n) = A000272(n) + A057500(n) = n^{n-2} + (n-1)(n-2)/2Sum_{r=1..n-2}( (n-3)!/(n-2-r)! )n^(n-2-r)
E.g.f.: log(1/(1-T(x)))/2 + T(x)/2 - 3*T(x)^2/4 + 1, where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Mar 23 2013
a(n) = ((n-1)*e^n*GAMMA(n-1,n)+n^(n-2)*(3-n))/2 for n>=1. - Peter Luschny, Jan 18 2016

Terms a(17) and beyond from Andrew Howroyd, Nov 07 2019

A137916 Number of n-node labeled graphs whose components are unicyclic graphs.

Original entry on oeis.org

1, 0, 0, 1, 15, 222, 3670, 68820, 1456875, 34506640, 906073524, 26154657270, 823808845585, 28129686128940, 1035350305641990, 40871383866109888, 1722832666898627865, 77242791668604946560, 3670690919234354407000, 184312149879830557190940, 9751080154504005703189791

Offset: 0

Washington Bomfim, Feb 22 2008

Also the number of labeled simple graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. The version without the choice condition is A116508, covering A367863. - Gus Wiseman, Jan 25 2024

			a(6) = 3670 because A057500(6) = 3660 and two triangles can be labeled in 10 ways.
From _Gus Wiseman_, Jan 25 2024: (Start)
The a(0) = 1 through a(4) = 15 simple graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}
                        {12,13,14,24}
                        {12,13,14,34}
                        {12,13,23,24}
                        {12,13,23,34}
                        {12,13,24,34}
                        {12,14,23,24}
                        {12,14,23,34}
                        {12,14,24,34}
                        {12,23,24,34}
                        {13,14,23,24}
                        {13,14,23,34}
                        {13,14,24,34}
                        {13,23,24,34}
                        {14,23,24,34}
(End)

V. F. Kolchin, Random Graphs. Encyclopedia of Mathematics and Its Applications 53. Cambridge Univ. Press, Cambridge, 1999.

Alois P. Heinz, Table of n, a(n) for n = 0..386 (terms n = 151..200 from Andrew Howroyd)
Eric Weisstein's World of Mathematics, Pseudoforest.
Wikipedia, Pseudoforest.

The connected case is A057500.
Row sums of A106239.
The unlabeled version is A137917.
Diagonal of A144228.
The version with loops appears to be A333331, unlabeled A368984.
Column k = 0 of A368924.
The complement is counted by A369143, unlabeled A369201, covering A369144.
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 simple graphs, covering A367869.
A140637 counts unlabeled non-choosable graphs, covering A369202.
A367867 counts non-choosable graphs, covering A367868.
Cf. A001434, A006649, A053530, A116508, A129271, A134964, A140638, A367862, A367863, A369191.

cy:= proc(n) option remember;
       binomial(n-1, 2)*add((n-3)!/(n-2-t)!*n^(n-2-t), t=1..n-2)
     end:
T:= proc(n,k) option remember; `if`(k=0, 1, `if`(k<0 or n T(n,n):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2008

nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Drop[Range[0, nn]! CoefficientList[Series[Exp[Log[1/(1 - t)]/2 - t/2 - t^2/4], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Jan 24 2012 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

A057500(p) = (p-1)! * p^p /2 * sum(k = 3,p, 1/(p^k*(p-k)!)); /* Vladeta Jovovic, A057500. */
F(n,N) = { my(s = 0, K, D, Mc); forpart(P = n, D = Set(P); K = vector(#D);
for(i=1, #D, K[i] = #select(x->x == D[i], Vec(P)));
Mc = n!/prod(i=1,#D, K[i]!);
s += Mc * prod(i = 1, #D, A057500(D[i])^K[i] / ( D[i]!^K[i])) , [3, n], [N, N]); s };
a(n)= {my(N); sum(N = 1, n, F(n,N))};

seq(n)={my(w=lambertw(-x+O(x*x^n))); Vec(serlaplace(exp(-log(1+w)/2 + w/2 - w^2/4)))} \\ Andrew Howroyd, May 18 2021

a(n) = Sum_{N = 1..n} ((n!/N!) * Sum_{n_1 + n_2 + ... + n_N = n} Product_{i = 1..N} ( A057500(n_i) / n_i! ) ). [V. F. Kolchin p. 31, (1.4.2)] replacing numerator terms n_i^(n_i-2) by A057500(n_i).
a(n) = A144228(n,n). - Alois P. Heinz, Sep 15 2008
E.g.f.: exp(B(T(x))) where B(x) = (log(1/(1-x))-x-x^2/2)/2 and T(x) is the e.g.f. for A000169 (labeled rooted trees). - Geoffrey Critzer, Jan 24 2012
a(n) ~ 2^(-1/4)*exp(-3/4)*GAMMA(3/4)*n^(n-1/4)/sqrt(Pi) * (1-7*Pi/(12*GAMMA(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Aug 16 2013
E.g.f.: exp(B(x)) where B(x) is the e.g.f. of A057500. - Andrew Howroyd, May 18 2021

a(0)=1 prepended by Andrew Howroyd, May 18 2021

A005703 Number of n-node connected graphs with at most one cycle.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 19, 44, 112, 287, 763, 2041, 5577, 15300, 42419, 118122, 330785, 929469, 2621272, 7411706, 21010378, 59682057, 169859257, 484234165, 1382567947, 3952860475, 11315775161, 32430737380, 93044797486, 267211342954, 768096496093, 2209772802169

Offset: 0

N. J. A. Sloane, R. K. Guy

a(n) is the number of pseudotrees on n nodes. - Eric W. Weisstein, Jun 11 2012

Also unlabeled connected graphs covering n vertices with at most n edges. For this definition we have a(1) = 0 and possibly a(0) = 0. - Gus Wiseman, Feb 20 2024

			From _Gus Wiseman_, Feb 20 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 8 graphs:
  {}  .  {12}  {12,13}     {12,13,14}     {12,13,14,15}
               {12,13,23}  {12,13,24}     {12,13,14,25}
                           {12,13,14,23}  {12,13,24,35}
                           {12,13,24,34}  {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}
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Washington Bomfim, Table of n, a(n) for n = 0..500
R. K. Guy, Letters to N. J. A. Sloane and J. W. Moon, 1988
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Pseudotree
Wikipedia, Pseudoforest

Cf. A000055, A000081, A001429 (labeled A057500), A134964 (number of pseudoforests, labeled A133686).
The labeled version is A129271.
The connected complement is A140636, labeled A140638.
Non-connected: A368834 (labeled A367869) or A370316 (labeled A369191).
A001187 counts connected graphs, unlabeled A001349.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A062734 counts connected graphs by number of edges.
Cf. A000272, A006649, A116508, A140637, A143543, A367862, A367863, A368951, A369197, A370317, A370318.

Needs["Combinatorica`"]; nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}];
a[0] = 0;
b = Drop[Flatten[
    sol = SolveAlways[
      0 == Series[
        t[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],
      x]; Table[a[n], {n, 0, nn}] /. sol], 1];
r[x_] := Sum[b[[n]] x^n, {n, 1, nn}]; c =
Drop[Table[
    CoefficientList[
     Series[CycleIndex[DihedralGroup[n], s] /.
       Table[s[i] -> r[x^i], {i, 1, n}], {x, 0, nn}], x], {n, 3,
     nn}] // Total, 1];
d[x_] := Sum[c[[n]] x^n, {n, 1, nn}]; CoefficientList[
Series[r[x] - (r[x]^2 - r[x^2])/2 + d[x] + 1, {x, 0, nn}], x] (* Geoffrey Critzer, Nov 17 2014 *)

\\ TreeGf gives gf of A000081.
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)}
seq(n)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec(1 + g(1) + (g(2) - g(1)^2)/2 + sum(k=3, n, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/k + if(k%2, g(1)*g(2)^(k\2), (g(1)^2+g(2))*g(2)^(k/2-1)/2))/2)}; \\ Andrew Howroyd and Washington Bomfim, May 15 2021

a(n) = A000055(n) + A001429(n).

More terms from Vladeta Jovovic, Apr 19 2000 and from Michael Somos, Apr 26 2000

a(27) corrected and a(28) and a(29) computed by Washington Bomfim, May 14 2008

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

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

Original entry on oeis.org

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.

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).

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.

cp's OEIS Frontend

A057500 Number of connected labeled graphs with n edges and n nodes.

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A129271 Number of labeled n-node connected graphs with at most one cycle.

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A137916 Number of n-node labeled graphs whose components are unicyclic graphs.

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A005703 Number of n-node connected graphs with at most one cycle.

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

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