A368951 -id:A368951 - 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

A014068 a(n) = binomial(n*(n+1)/2, n).

Original entry on oeis.org

1, 1, 3, 20, 210, 3003, 54264, 1184040, 30260340, 886163135, 29248649430, 1074082795968, 43430966148115, 1917283000904460, 91748617512913200, 4730523156632595024, 261429178502421685800, 15415916972482007401455, 966121413245991846673830, 64123483527473864490450300

Offset: 0

N. J. A. Sloane

nonn

Product of next n numbers divided by product of first n numbers. E.g., a(4) = (7*8*9*10)/(1*2*3*4)= 210. - Amarnath Murthy, Mar 22 2004

Also the number of labeled loop-graphs with n vertices and n edges. The covering case is A368597. - Gus Wiseman, Jan 25 2024

			From _Gus Wiseman_, Jan 25 2024: (Start)
The a(0) = 1 through a(3) = 20 loop-graph edge-sets (loops shown as singletons):
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
             {{1},{1,2}}  {{1},{2},{1,2}}
             {{2},{1,2}}  {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{1,3}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{2},{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}}
(End)

Harvey P. Dale, Table of n, a(n) for n = 0..370
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A000217, A022915, A135860, A135861, A135862.
Diagonal of A084546.
Without loops we have A116508, covering A367863, unlabeled A006649.
Allowing edges of any positive size gives A136556, covering A054780.
The covering case is A368597.
The unlabeled version is A368598, covering A368599.
The connected case is A368951.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 (shifted left) counts loop-graphs, covering A322661.
A006129 counts covering simple graphs, connected A001187.
A058891 counts set-systems, unlabeled A000612.
Cf. A000085, A000272, A057500, A333331, A368596, A368730.

[Binomial(Binomial(n+1,2), n): n in [0..40]]; // G. C. Greubel, Feb 19 2022

Binomial[First[#],Last[#]]&/@With[{nn=20},Thread[{Accumulate[ Range[ 0,nn]], Range[ 0,nn]}]] (* Harvey P. Dale, May 27 2014 *)

from math import comb
def A014068(n): return comb(comb(n+1,2),n) # Chai Wah Wu, Jul 14 2024

[(binomial(binomial(n+1, n-1), n)) for n in range(20)] # Zerinvary Lajos, Nov 30 2009

For n >= 1, Product_{k=1..n} a(k) = A022915(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
For n > 0, a(n) = A022915(n)/A022915(n-1). - Gerald McGarvey, Jul 26 2004
a(n) = binomial(T(n+1), T(n)) where T(n) = the n-th triangular number. - Amarnath Murthy, Jul 14 2005
a(n) = binomial(binomial(n+2, n), n+1) for n >= -1. - Zerinvary Lajos, Nov 30 2009
From Peter Bala, Feb 27 2020: (Start)
a(p) == (p + 1)/2 ( mod p^3 ) for prime p >= 5 (apply Mestrovic, equation 37).
Conjectural: a(2*p) == p*(2*p + 1) ( mod p^4 ) for prime p >= 5. (End)
a(n) = A084546(n,n). - Gus Wiseman, Jan 25 2024
a(n) = [x^n] (1+x)^(n*(n+1)/2). - Vaclav Kotesovec, Aug 06 2025

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

A368597 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.

Original entry on oeis.org

1, 1, 3, 17, 150, 1803, 27364, 501015, 10736010, 263461265, 7283725704, 223967628066, 7581128184175, 280103206674480, 11216492736563655, 483875783716549277, 22371631078155742764, 1103548801569848115255, 57849356643299101021960, 3211439288584038922502820

Offset: 0

Gus Wiseman, Jan 04 2024

nonn

It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.

			The a(0) = 1 through a(3) = 17 set-systems:
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
             {{1},{1,2}}  {{1},{2},{1,3}}
             {{2},{1,2}}  {{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
Eric Weisstein's World of Mathematics, Graph Loop.

This is the covering case of A014068.
Allowing edges of any positive size gives A054780, covering case of A136556.
Allowing any number of edges gives A322661, connected A062740.
The case of just pairs is A367863, covering case of A116508.
The unlabeled version is A368599.
The version contradicting strict AOC is A368730.
The connected case is A368951.
A000085 counts set partitions into singletons or pairs.
A006129 counts covering graphs, connected A001187.
A058891 counts set-systems, unlabeled A000612.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
Cf. A000272, A000666, A057500, A066383, A333331, A367869, A368596, A368600, A368928, A368951, A369199.

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

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n)) \\ Andrew Howroyd, Jan 06 2024

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n). - Andrew Howroyd, Jan 06 2024

Terms a(7) and beyond from Andrew Howroyd, Jan 06 2024

A333331 Number of integer points in the convex hull in R^n of parking functions of length n.

Original entry on oeis.org

1, 3, 17, 144, 1623, 22804, 383415, 7501422

Offset: 1

Richard Stanley, Mar 15 2020

It is observed by Gus Wiseman in A368596 and A368730 that this sequence appears to be the complement of those sequences. If this is the case, then a(n) is the number of labeled graphs with loops allowed in which each connected component has an equal number of vertices and edges and the conjectured formula holds. Terms for n >= 9 are expected to be 167341283, 4191140394, 116425416531, ... - Andrew Howroyd, Jan 10 2024
From Gus Wiseman, Mar 22 2024: (Start)
An equivalent conjecture is that a(n) is the number of loop-graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. I call these graphs choosable. For example, the a(3) = 17 choosable loop-graphs are the following (loops shown as singletons):
  {{1},{2},{3}}  {{1},{2},{1,3}}  {{1},{1,2},{1,3}}  {{1,2},{1,3},{2,3}}
                 {{1},{2},{2,3}}  {{1},{1,2},{2,3}}
                 {{1},{3},{1,2}}  {{1},{1,3},{2,3}}
                 {{1},{3},{2,3}}  {{2},{1,2},{1,3}}
                 {{2},{3},{1,2}}  {{2},{1,2},{2,3}}
                 {{2},{3},{1,3}}  {{2},{1,3},{2,3}}
                                  {{3},{1,2},{1,3}}
                                  {{3},{1,2},{2,3}}
                                  {{3},{1,3},{2,3}}
(End)

			For n=2 the parking functions are (1,1), (1,2), (2,1). They are the only integer points in their convex hull. For n=3, in addition to the 16 parking functions, there is the additional point (2,2,2).

R. P. Stanley (Proposer), Problem 12191, Amer. Math. Monthly, 127:6 (2020), 563.

Thomas Selig, The stochastic sandpile model on complete graphs, arXiv:2209.07301 [math.PR], 2022.

All of the following relative references pertain to the conjecture:
The case of unique choice A000272.
The version without the choice condition is A014068, covering A368597.
The case of just pairs A137916.
For any number of edges of any positive size we have A367902.
The complement A368596, covering A368730.
Allowing edges of any positive size gives A368601, complement A368600.
Counting by singletons gives A368924.
For any number of edges we have A368927, complement A369141.
The connected case is A368951.
The unlabeled version is A368984, complement A368835.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
Cf. A000169, A000666, A057500, A062740, A129271, A133686, A367863, A367903, A369146, A369199.

Conjectured e.g.f.: exp(-log(1-T(x))/2 + T(x)/2 - T(x)^2/4) where T(x) = -LambertW(-x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 10 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

A368983 Number of connected graphs with loops (symmetric relations) on n unlabeled vertices with n edges.

Original entry on oeis.org

1, 1, 1, 3, 6, 14, 33, 81, 204, 526, 1376, 3648, 9792, 26485, 72233, 198192, 546846, 1515687, 4218564, 11782427, 33013541, 92759384, 261290682, 737688946, 2086993034, 5915398230, 16795618221, 47763406249, 136028420723, 387928330677, 1107692471888, 3166613486137

Offset: 0

Andrew Howroyd, Jan 11 2024

nonn

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

			Representatives of the a(3) = 3 graphs are:
   {{1,2}, {1,3}, {2,3}},
   {{1}, {1,2}, {1,3}},
   {{1}, {1,2}, {2,3}}.

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

A diagonal of A322114.
The labeled version is A368951.
Cf. A000081, A001429, A027852, A068051, A283755, A368984 (not necessarily connected).

\\ 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 + (sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2 - (g(1)^2 + g(2)))/2)}

a(n) = A000081(n) + A001429(n) = A068051(n) - A027852(n) for n > 0.

A368849 Triangle read by rows: T(n, k) = binomial(n, k - 1)(k - 1)^(k - 1)(n - k)*(n - k + 1)^(n - k).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 18, 6, 0, 0, 192, 72, 48, 0, 0, 2500, 960, 720, 540, 0, 0, 38880, 15000, 11520, 9720, 7680, 0, 0, 705894, 272160, 210000, 181440, 161280, 131250, 0, 0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0

Offset: 0

Peter Luschny, Jan 11 2024

A motivation for this triangle was to provide an alternative sum representation for A001864(n) = n! * Sum_{k=0..n-2} n^k/k!. See formula 3 and formula 15 in Riordan and Sloane.

			Triangle starts:
  [0] [0]
  [1] [0,        0]
  [2] [0,        2,       0]
  [3] [0,       18,       6,       0]
  [4] [0,      192,      72,      48,      0]
  [5] [0,     2500,     960,     720,     540,       0]
  [6] [0,    38880,   15000,   11520,    9720,    7680,       0]
  [7] [0,   705894,  272160,  210000,  181440,  161280,  131250,       0]
  [8] [0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0]

John Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.

T(n, 1) = A066274(n) for n >= 1.
T(n, 1)/(n - 1) = A000169(n) for n >= 2.
T(n, n - 1) = 2*A081133(n) for n >= 1.
Sum_{k=0..n} T(n, k) = A001864(n).
(Sum_{k=0..n} T(n, k)) / n = A000435(n) for n >= 1.
(Sum_{k=0..n} T(n, k)) * n / 2 = A262973(n) for n >= 1.
(Sum_{k=2..n} T(n, k)) / (2*n) = A057500(n) for n >= 1.
T(n, 1)/(n - 1) + (Sum_{k=2..n} T(n, k)) / (2*n) = A368951(n) for n >= 2.
Sum_{k=0..n} (-1)^(k-1) * T(n, k) = A368981(n).

A368849[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k);
Table[A368849[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2024 *)

def T(n, k):
    return binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k)
for n in range(0, 9): print([n], [T(n, k) for k in range(n + 1)])

A370318 Number of labeled simple graphs with n vertices and the same number of edges as covered vertices, such that the edge set is connected.

Original entry on oeis.org

0, 0, 0, 1, 19, 307, 5237, 99137, 2098946, 49504458, 1291570014, 37002273654, 1156078150969, 39147186978685, 1428799530304243, 55933568895261791, 2338378885159906196, 103995520598384132516, 4903038902046860966220, 244294315694676224001852, 12827355456239840407125363

Offset: 0

Gus Wiseman, Feb 18 2024

nonn

The case of an empty edge set is excluded.

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

The covering case is A057500, which is also the covering case of A370317.
This is the connected case of A367862, covering A367863.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A062734 counts connected graphs by edge count.
A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
A143543 counts simple labeled graphs by number of connected components.
A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
Cf. A001429, A006649, A061540, A116508, A323818, A367916, A368951, A369197.

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}]],Length[#]==Length[Union@@#] && Length[csm[#]]==1&]],{n,0,5}]

\\ Compare A370317; use A057500 for efficiency.
a(n)=n!*polcoef(polcoef(exp(x*y + O(x*x^n))*(-x+log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2)*x^k/k!, O(x*x^n)))), n), n) \\ Andrew Howroyd, Feb 19 2024

Binomial transform of A057500 (if the null graph is not connected).

a(n) = n!*[x^n][y^n] exp(x*y)*(-x + log(Sum_{k>=0} (1 + y)^binomial(k, 2)*x^k/k!)). - Andrew Howroyd, Feb 19 2024

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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A014068 a(n) = binomial(n*(n+1)/2, n).

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

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A368597 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.

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A333331 Number of integer points in the convex hull in R^n of parking functions of length n.

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A368849 Triangle read by rows: T(n, k) = binomial(n, k - 1)(k - 1)^(k - 1)(n - k)*(n - k + 1)^(n - k).