A062734 -id:A062734 - cp's OEIS frontend

A001187 Number of connected labeled graphs with n nodes.

1, 1, 1, 4, 38, 728, 26704, 1866256, 251548592, 66296291072, 34496488594816, 35641657548953344, 73354596206766622208, 301272202649664088951808, 2471648811030443735290891264, 40527680937730480234609755344896, 1328578958335783201008338986845427712

Offset: 0

N. J. A. Sloane

"Based on experimental data obtained using the software LattE [14] and the Online Encyclopedia of Integer Sequences [19], we make the following conjecture: Conjecture 11. For j >= 2, Vol(C_j ) is equal to the number of labeled connected graphs on j - 1 vertices." [Beck et al., 2011]
For n > 1, a(n) is log-convex. Furthermore, a(n+1)*a(n-1) ~ 2*a(n)*a(n). - Ran Pan, Oct 28 2015
a(n) is also the number of tournaments on {1,...,n} for which 1 is reachable from every vertex. - Don Knuth, Aug 06 2020

			E.g.f.: 1 + x + x^2/2! + 4*x^3/3! + 38*x^4/4! + 728*x^5/5! + 26704*x^6/6! + 1866256*x^7/7! + 251548592*x^8/8! + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 398-402.
D. G. Cantor, personal communication.
Cowan, D. D.; Mullin, R. C.; Stanton, R. G. Counting algorithms for connected labelled graphs. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), pp. 225-236. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975. MR0414417 (54 #2519).
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 518.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 7.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.1.
H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 78.

T. D. Noe, Table of n, a(n) for n = 0..50
Matthias Beck, Benjamin Braun and Nguyen Le, Mahonian partition identities via polyhedral geometry, arXiv:1103.1070 [math.NT], 2011.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, MS Thesis, 2002.
Marco Coraggio and Mario di Bernardo, Data-driven design of complex network structures to promote synchronization, arXiv:2309.10941 [eess.SY], 2023.
Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 138.
Philippe Fournier Viger, Ganghuan He, Chun-Wei Jerry Lin, and Heitor Murilo Gomes, Mining Attribute Evolution Rules in Dynamic Attributed Graphs, Proceedings of the Big Data Analytics and Knowledge Discovery, 22nd International Conference, (Bratislava, Slovakia, DaWaK 2020).
Adriano M. Garsia, James Haglund, Dun Qiu, and Marino Romero, e-Positivity Results and Conjectures, arXiv:1904.07912 [math.CO], 2019.
E. N. Gilbert, Enumeration of labelled graphs, Canad. J. Math., 8 (1956), 405-411.
E. N. Gilbert, Enumeration of labelled graphs (Annotated scanned copy)
Markus Kirchweger and Stefan Szeider, SAT Modulo Symmetries for Graph Generation and Enumeration, ACM Trans. Comput. Logic (2024). See p. 27.
M. Konvalinka and I. Pak, Cayley compositions, partitions, polytopes, and geometric bijections, preprint.
M. Konvalinka and I. Pak, Cayley compositions, partitions, polytopes, and geometric bijections, Journal of Combinatorial Theory, Series A, Volume 123, Issue 1, April 2014, Pages 86-91.
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See p. 32.
Arun P. Mani and R. J. Stones, The Number of Labeled Connected Graphs Modulo Prime Powers, SIAM Journal on Discrete Mathematics, Vol. 30, No. 2, pp. 1046-1057.
Alexey A. Melnikov, Leonid E. Fedichkin, Ray-Kuang Lee, and Alexander Alodjants, Machine learning transfer efficiencies for noisy quantum walks, arXiv:2001.05472 [quant-ph], 2020. Also in Advanced Quantum Technologies (2020) Vol. 3, 1900115.
Albert Nijenhuis and Herbert S. Wilf, The enumeration of connected graphs and linked diagrams, J. Combin. Theory Ser. A 27 (1979), no. 3, 356--359. MR0555804 (82b:05074)
J. Novak, Three lectures on free probability, arXiv preprint arXiv:1205.2097 [math.CO], 2012.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
R. W. Robinson, First 50 terms of A1187 and A1188
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Labeled Graph
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 87, Eq. 3.10.2.

Logarithmic transform of A006125 (labeled graphs).
Row sums of triangle A062734.
Cf. A053549.

m:=30;
f:= func< x | 1+Log( (&+[2^Binomial(n,2)*x^n/Factorial(n): n in [0..m+3]]) ) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Oct 04 2022

t1 := 1+log( add(2^binomial(n,2)*x^n/n!,n=0..30)): t2 := series(t1,x,30): A001187 := n->n!*coeff(t2,x,n);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
      add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*a(k), k=1..n-1)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 26 2013
# Alternative:
a := proc(n) option remember;
    2^((n-1)*n/2) - add(binomial(n-1, k)*2^((k-n+1)*(k-n+2)/2)*a(k+1), k=0..n-2)
end:
seq(a(n), n=0..16); # Peter Luschny, Feb 21 2023

m:=20; g = Sum[2^Binomial[n, 2] x^n/n!, {n,0,m}]; Range[0,m]! CoefficientList[Series[Log[g] +1, {x,0,m}], x] (* Geoffrey Critzer, Nov 12 2011 *)
a[n_]:= a[n]= If[n==0, 1, 2^(n*(n-1)/2) - Sum[k*Binomial[n, k]* 2^((n-k)*(n-k-1)/2)*a[k], {k,1,n-1}]/n]; Table[a[n], {n,0,20}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[1 +Log[ Sum[2^(k(k-1)/2) x^k/k!, {k,0, n}]], {x, 0, n}]]; (* Michael Somos, Jul 11 2019 *)

{a(n) = if(n<0, 0, n! * polcoeff( 1 + log( sum( k=0, n, 2^binomial(k, 2) * x^k / k!, x * O(x^n))), n))}; /* Michael Somos, Jun 12 2000 */

from functools import lru_cache
import gmpy2
@lru_cache(None)
def A001187(n):
  if n == 0:
    return 1
  s = gmpy2.mpz(0)
  for k in range(1, n):
    s += k * gmpy2.comb(n, k) * 2**((n - k)*(n - k - 1)//2) * A001187(k)
  return 2**(n*(n-1)//2) - s // n # John Reimer Morales, Aug 15 2025

@cached_function
def A001187(n):
    if n == 0: return 1
    return 2^(n*(n-1)/2)- sum(k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*A001187(k) for k in (1..n-1))/n
[A001187(n) for n in (0..15)] # Peter Luschny, Jan 17 2016

n * 2^binomial(n, 2) = Sum_{k=1..n} binomial(n, k) * k * a(k) * 2^binomial(n-k, 2).

E.g.f.: 1 + log(Sum_{n>=0} 2^binomial(n, 2) * x^n / n!). - Michael Somos, Jun 12 2000

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

Original entry on oeis.org

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

A054548 Triangular array giving number of labeled graphs on n unisolated nodes and k=0...n*(n-1)/2 edges.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 3, 1, 0, 0, 3, 16, 15, 6, 1, 0, 0, 0, 30, 135, 222, 205, 120, 45, 10, 1, 0, 0, 0, 15, 330, 1581, 3760, 5715, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 0, 0, 0, 0, 315, 4410, 23604, 73755, 159390, 259105, 331716, 343161, 290745, 202755, 116175

Offset: 0

Vladeta Jovovic, Apr 09 2000

			From _Gus Wiseman_, Feb 14 2024: (Start)
Triangle begins:
   1
   0
   0   1
   0   0   3   1
   0   0   3  16  15   6   1
   0   0   0  30 135 222 205 120  45  10   1
Row n = 4 counts the following graphs:
  .  .  12-34  12-13-14  12-13-14-23  12-13-14-23-24  12-13-14-23-24-34
        13-24  12-13-24  12-13-14-24  12-13-14-23-34
        14-23  12-13-34  12-13-14-34  12-13-14-24-34
               12-14-23  12-13-23-24  12-13-23-24-34
               12-14-34  12-13-23-34  12-14-23-24-34
               12-23-24  12-13-24-34  13-14-23-24-34
               12-23-34  12-14-23-24
               12-24-34  12-14-23-34
               13-14-23  12-14-24-34
               13-14-24  12-23-24-34
               13-23-24  13-14-23-24
               13-23-34  13-14-23-34
               13-24-34  13-14-24-34
               14-23-24  13-23-24-34
               14-23-34  14-23-24-34
               14-24-34
(End)

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, Page 29, Exercise 1.4.

Michael De Vlieger, Table of n, a(n) for n = 0..10700
A. N. Bhavale and B. N. Waphare, Basic retracts and counting of lattices, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99.
Ashok Nivrutti Bhavale, Equivalence of labeled graphs and lattices, arXiv:2501.05064 [math.CO], 2025. See pp. 1-2, 13.
R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4.

Row sums give A006129. Cf. A054547.
The connected case is A062734, with loops A369195.
This is the covering case of A084546.
Column sums are A121251, with loops A173219.
The version with loops is A369199, row sums A322661.
The unlabeled version is A370167, row sums A002494.
A006125 counts simple graphs; also loop-graphs if shifted left.
Cf. A000666, A006649, A062740, A066383, A121252, A133686, A322700, A368597, A369196, A369197.

nn=5; s=Sum[(1+y)^Binomial[n,2]  x^n/n!, {n,0,nn}]; Range[0,nn]! CoefficientList[Series[ s Exp[-x], {x,0,nn}], {x,y}] //Grid  (* returns triangle indexed at n = 0, Geoffrey Critzer, Oct 07 2012 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}],{k}],Union@@#==Range[n]&]],{n,0,5},{k,0,Binomial[n,2]}] (* Gus Wiseman, Feb 14 2024 *)

T(n, k) = Sum_{i=0..n} (-1)^(n-i)*C(n, i)*C(C(i, 2), k), k=0...n*(n-1)/2.

E.g.f.: exp(-x)*Sum_{n>=0} (1 + y)^C(n,2)*x^n/n!. - Geoffrey Critzer, Oct 07 2012

a(0) prepended by Gus Wiseman, Feb 14 2024

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

Original entry on oeis.org

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

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

A372170 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k triangles, 0 <= k <= binomial(n,3).

Original entry on oeis.org

1, 1, 2, 7, 1, 41, 16, 6, 0, 1, 388, 290, 195, 70, 40, 30, 0, 10, 0, 0, 1, 5789, 6980, 6910, 4560, 3030, 2292, 1230, 780, 600, 180, 236, 60, 45, 60, 0, 0, 15, 0, 0, 0, 1, 133501, 235270, 313705, 302505, 260890, 222509, 174615, 126780, 102970, 67165, 50134, 37485, 20370, 17990, 11445, 6552, 4515, 3570, 1680, 1785, 154, 735, 455, 140, 0, 105, 105, 0, 0, 0, 21, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Apr 23 2024

			Triangle begins:
     1
     1
     2
     7    1
    41   16    6    0    1
   388  290  195   70   40   30    0   10    0    0    1
   ...
For example, the T(4,1) = 16 graphs are:
  12-13-23
  12-14-24
  13-14-34
  23-24-34
  12-13-14-23
  12-13-14-24
  12-13-14-34
  12-13-23-24
  12-13-23-34
  12-14-23-24
  12-14-24-34
  12-23-24-34
  13-14-23-34
  13-14-24-34
  13-23-24-34
  14-23-24-34

Andrew Howroyd, Table of n, a(n) for n = 0..340 (rows 0..10)

Row sums are A006125, covering A006129.
Row lengths are A050407.
Counting edges instead of triangles gives A084546, covering A054548.
Column k = 0 is A213434, covering A372168.
The unlabeled version is A263340.
The covering case is A372167, unlabeled A372173.
Column k = 1 is A372172, covering A372171.
For all cycles (not just triangles) we have A372176, covering A372175.
A001858 counts acyclic graphs, unlabeled A005195.
A367867 counts non-choosable graphs, covering A367868.
A372193 counts unicyclic graphs, unlabeled A236570, covering A372191.
Cf. A000088, A000272, A002494, A006785, A053530, A062734, A121251, A372194.

cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}]&&MemberQ[y,{#[[1]],#[[3]]}]&&MemberQ[y,{#[[2]],#[[3]]}]&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[cys[#]]==k&]],{n,0,5},{k,0,Binomial[n,3]}]

Binomial transform of columns of A372167.

a(42) onwards from Andrew Howroyd, Dec 29 2024

A343088 Triangle read by rows: T(n,k) is the number of connected labeled graphs with n edges and k vertices, 1 <= k <= n+1.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 0, 0, 1, 16, 0, 0, 0, 15, 125, 0, 0, 0, 6, 222, 1296, 0, 0, 0, 1, 205, 3660, 16807, 0, 0, 0, 0, 120, 5700, 68295, 262144, 0, 0, 0, 0, 45, 6165, 156555, 1436568, 4782969, 0, 0, 0, 0, 10, 4945, 258125, 4483360, 33779340, 100000000

Offset: 0

Andrew Howroyd, Apr 14 2021

			Triangle begins:
  1;
  0, 1;
  0, 0, 3;
  0, 0, 1, 16;
  0, 0, 0, 15, 125;
  0, 0, 0,  6, 222, 1296;
  0, 0, 0,  1, 205, 3660,  16807;
  0, 0, 0,  0, 120, 5700,  68295,  262144;
  0, 0, 0,  0,  45, 6165, 156555, 1436568, 4782969;
  ...

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

Main diagonal is A000272.
Subsequent diagonals give the number of connected labeled graphs with n nodes and n+k edges for k=0..11: A057500, A061540, A061541, A061542, A061543, A096117, A061544 A096150, A096224, A182294, A182295, A182371.
Row sums are A322137.
Column sums are A001187.
Cf. A054923 (unlabeled), A062734 (transpose), A290776 (multigraphs), A322147 (loops allowed), A331437 (series-reduced).

row[n_] := (SeriesCoefficient[#, {y, 0, n}]& /@ CoefficientList[ Log[Sum[x^k*(1+y)^Binomial[k, 2]/k!, {k, 0, n+1}]] + O[x]^(n+2), x]* Range[0, n+1]!) // Rest;
Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Aug 03 2022, after Andrew Howroyd *)

Row(n)={Vec(serlaplace(polcoef(log(O(x^2*x^n)+sum(k=0, n+1, x^k*(1 + y + O(y*y^n))^binomial(k, 2)/k!)), n, y)), -(n+1))}
{ for(n=0, 8, print(Row(n))) }

A370167 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with k = 0..binomial(n,2) edges.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 0, 0, 1, 4, 5, 5, 4, 2, 1, 1, 0, 0, 0, 1, 3, 9, 15, 20, 22, 20, 14, 9, 5, 2, 1, 1, 0, 0, 0, 0, 1, 6, 20, 41, 73, 110, 133, 139, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1, 0, 0, 0, 0, 1, 3, 15, 50, 124, 271, 515, 832, 1181, 1460, 1581, 1516, 1291, 970, 658, 400, 220, 114, 56, 24, 11, 5, 2, 1, 1

Offset: 0

Gus Wiseman, Feb 15 2024

			Triangle begins:
  1
  0
  0  1
  0  0  1  1
  0  0  1  2  2  1  1
  0  0  0  1  4  5  5  4  2  1  1
  0  0  0  1  3  9 15 20 22 20 14  9  5  2  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..1350 (rows 0..20)

Column sums are A000664.
Row sums are A002494.
This is the covering case of A008406, labeled A084546.
The labeled version is A054548, row sums A006129, column sums A121251.
The connected case is A054924, row sums A001349, column sums A002905.
The labeled connected case is A062734, with loops A369195.
The connected case with loops is A283755, row sums A054921.
The labeled version w/ loops is A369199, row sums A322661, col sums A173219.
Cf. A000666, A006125, A006649, A054547, A066383, A322700.

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],{2}],{k}],Union@@#==Range[n]&]]], {n,0,5},{k,0,Binomial[n,2]}]

\\ G(n) defined in A008406.
row(n)={Vecrev(G(n)-if(n>0, G(n-1)), binomial(n,2)+1)}
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Feb 19 2024

a(42) onwards from Andrew Howroyd, Feb 19 2024

A123527 Triangular array T(n,k) giving number of connected graphs with n labeled nodes and k edges (n >= 1, n-1 <= k <= n(n-1)/2).

Original entry on oeis.org

1, 1, 3, 1, 16, 15, 6, 1, 125, 222, 205, 120, 45, 10, 1, 1296, 3660, 5700, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 16807, 68295, 156555, 258125, 331506, 343140, 290745, 202755, 116175, 54257, 20349, 5985, 1330, 210, 21, 1, 262144, 1436568

Offset: 1

N. J. A. Sloane, Nov 13 2006

			Triangle begins:
n = 1
  k = 0: 1
  ****** total(1) = 1
n = 2
  k = 1: 1
  ****** total(2) = 1
n = 3
  k = 2: 3
  k = 3: 1
  ****** total(3) = 4
n = 4
  k = 3: 16
  k = 4: 15
  k = 5:  6
  k = 6:  1
  ****** total(4) = 38
n = 5
  k = 4: 125
  k = 5: 222
  k = 6: 205
  k = 7: 120
  k = 8:  45
  k = 9:  10
  k = 10:  1
  ****** total(5) = 728

Cowan, D. D.; Mullin, R. C.; Stanton, R. G. Counting algorithms for connected labelled graphs. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), pp. 225-236. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975. MR0414417 (54 #2519). - From N. J. A. Sloane, Apr 06 2012
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.

Seiichi Manyama, Rows n = 1..40, flattened (rows 1..14 from R. W. Robinson, rows 15..20 from Henrique G. G. Pereira)
T. Yanagita, T. Ichinomiya, Thermodynamic Characterization of Synchronization-Optimized Oscillator-Networks, arXiv preprint arXiv:1409.1979 [nlin.AO], 2014.

See A062734 for another version with more information. Row sums give A001187.

nn = 8; a = Sum[(1 + y)^Binomial[n, 2] x^n/n!, {n, 0, nn}]; f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Drop[Range[0, nn]! CoefficientList[Series[Log[a], {x, 0, nn}], {x, y}],1]]] (* Geoffrey Critzer, Dec 08 2011 *)
T[ n_, k_] := If[ n < 1, 0, Coefficient[ n! SeriesCoefficient[ Log[ Sum[ (1 + y)^Binomial[m, 2] x^m/m!, {m, 0, n}]], {x, 0, n}], y, n - 1 + k]]; (* Michael Somos, Aug 12 2017 *)

For k >= C(n-1, 2) + 1 (not smaller!), T(n,k) = C(C(n,2),k) where C(n,k) is the binomial coefficient. See A084546. - Geoffrey Critzer, Dec 08 2011

cp's OEIS Frontend

A001187 Number of connected labeled graphs with n nodes.

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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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A054548 Triangular array giving number of labeled graphs on n unisolated nodes and k=0...n*(n-1)/2 edges.

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

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