A002905 -id:A002905 - cp's OEIS frontend

A008406 Triangle T(n,k) read by rows, giving number of graphs with n nodes (n >= 1) and k edges (0 <= k <= n(n-1)/2).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 4, 6, 6, 6, 4, 2, 1, 1, 1, 1, 2, 5, 9, 15, 21, 24, 24, 21, 15, 9, 5, 2, 1, 1, 1, 1, 2, 5, 10, 21, 41, 65, 97, 131, 148, 148, 131, 97, 65, 41, 21, 10, 5, 2, 1, 1, 1, 1, 2, 5, 11, 24, 56, 115, 221, 402, 663, 980, 1312, 1557, 1646, 1557

Offset: 1

N. J. A. Sloane, Mar 15 1996

T(n,k)=1 for n>=2 with k=0, k=1, k=n*(n-1)/2-1 and k=n*(n-1)/2 (therefore the quadruple {1,1,1,1} marks the transition to the next sublist for a given number of vertices (n>2)). [Edited by Peter Munn, Mar 20 2021]

			Triangle begins:
1,
1,1,
1,1,1,1,
1,1,2,3,2,1,1, [graphs with 4 nodes and from 0 to 6 edges]
1,1,2,4,6,6,6,4,2,1,1,
1,1,2,5,9,15,21,24,24,21,15,9,5,2,1,1,
1,1,2,5,10,21,41,65,97,131,148,148,131,97,65,41,21,10,5,2,1,1,
...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 264.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 519.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 214.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 240.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

R. W. Robinson, Rows 1 to 20 of triangle, flattened
Leonid Bedratyuk, A new formula for the generating function of the numbers of simple graphs, arXiv:1512.06355 [math.CO], 2015.
FindStat - Combinatorial Statistic Finder, The number of edges of a graph.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) Table 65.
Sriram V. Pemmaraju, Combinatorica 2.0
Marko R. Riedel, Number of distinct connected digraphs
Gordon Royle, Small graphs
S. S. Skiena, Generating graphs
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Overview of the 12 Parts (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 1
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 2
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 3
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 4
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 5
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 6
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 7
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 8
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 9
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 10
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 11
Peter Steinbach, Field Guide to Simple Graphs, Volume 2, Part 12
James Turner and William H. Kautz, A survey of progress in graph theory in the Soviet Union SIAM Rev. 12 1970 suppl. iv+68 pp. MR0268074 (42 #2973). See p. 19.
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Simple Graph
A. E. Yurtsun, Principles of enumeration of the number of graphs, Ukrainian Mathematical Journal, January-February, 1967, Volume 19, Issue 1, pp 123-125, DOI 10.1007/BF01085184.

Row sums give A000088.
Cf. A046742, A046751, A000717, A001432, A001431, A001430, A001433, A001434, A048179, A048180 etc.
Cf. also A039735, A002905, A054924 (connected), A084546 (labeled graphs).
Row lengths: A000124; number of connected graphs for given number of vertices: A001349; number of graphs for given number of edges: A000664.
Cf. also A000055.

seq(seq(GraphTheory:-NonIsomorphicGraphs(v,e),e=0..v*(v-1)/2),v=1..9); # Robert Israel, Dec 22 2015

<< Combinatorica`; Table[CoefficientList[GraphPolynomial[n, x], x], {n, 8}] // Flatten (* Eric W. Weisstein, Mar 20 2013 *)
<< Combinatorica`; Table[NumberOfGraphs[v, e], {v, 8}, {e, 0, Binomial[v, 2]}] // Flatten (* Eric W. Weisstein, May 17 2017 *)
permcount[v_] := Module[{m=1, s=0, k=0, t}, For[i=1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
edges[v_, t_] := Product[Product[g = GCD[v[[i]], v[[j]]]; t[v[[i]]*v[[j]]/ g]^g,{j, 1, i-1}], {i, 2, Length[v]}]*Product[c = v[[i]]; t[c]^Quotient[ c-1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
row[n_] := Module[{s = 0}, Do[s += permcount[p]*edges[p, 1 + x^#&], {p, IntegerPartitions[n]}]; s/n!] // Expand // CoefficientList[#, x]&;
Array[row, 8] // Flatten (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
G(n, A=0) = {my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i+A)); s/n!}
{ for(n=1, 7, print(Vecrev(G(n)))) } \\ Andrew Howroyd, Oct 22 2019, updated  Jan 09 2024

def T(n,k):
    return len(list(graphs(n, size=k)))
# Ralf Stephan, May 30 2014

O.g.f. for n-th row: 1/n! Sum_g det(1-g z^2)/det(1-g z) where g runs through the natural matrix representation of the pair group A^2_n (for A^2_n see F. Harary and E. M. Palmer, Graphical Enumeration, page 83). - Leonid Bedratyuk, Sep 23 2014

Additional comments from Arne Ring (arne.ring(AT)epost.de), Oct 03 2002

Text belonging in a different sequence deleted by Peter Munn, Mar 20 2021

A000664 Number of graphs with n edges.

Original entry on oeis.org

1, 1, 2, 5, 11, 26, 68, 177, 497, 1476, 4613, 15216, 52944, 193367, 740226, 2960520, 12334829, 53394755, 239544624, 1111261697, 5320103252, 26237509076, 133087001869, 693339241737, 3705135967663, 20286965943329, 113694201046379, 651571521170323, 3815204365835840, 22806847476040913, 139088381010541237, 864777487052916454

Offset: 0

N. J. A. Sloane

These are simple graphs, unlabeled, with no isolated nodes, but are not necessarily connected.

			n=1: o-o (1)
n=2: o-o o-o, o-o-o (2)
n=3: o-o o-o o-o, o-o-o o-o, o-o-o-o, Y, triangle (5)
n=4: o-o o-o o-o o-o, o-o-o o-o o-o, o-o-o o-o-o, o-o o-o-o-o, o-o Y, o-o triangle,
o-o-o-o-o, >o-o-o, ><, square, triangle with tail (11)

W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Max Alekseyev, Table of n, a(n) for n = 0..60
Nicolas Borie, The Hopf Algebra of graph invariants, arXiv preprint arXiv:1511.05843 [math.CO], 2015.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Fran Herr and Legrand Jones II, Iterated Jump Graphs, arXiv:2205.01796 [math.CO], 2022.
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Overview of the 11 Parts (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 1
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 2
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 3
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 4
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 5
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 6
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 7
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 8
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 9
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 10
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 11

Cf. A002905, A008406, A053418.
Row sums of A275421.
Cf. also A000088, A000055.

<< Combinatorica`; Table[NumberOfGraphs[2 n, n], {n, 0, 10}] (* Eric W. Weisstein, Oct 30 2017 *)
<< Combinatorica`; Table[Coefficient[GraphPolynomial[2 n, x], x, n], {n, 0, 10}] (* Eric W. Weisstein, Oct 30 2017 *)

a(n) = A008406(2*n,n). - Max Alekseyev, Sep 13 2016

Euler transform of A002905 (ignoring A002905(0)). - Franklin T. Adams-Watters Jul 03 2009

More terms from Vladeta Jovovic, Jan 08 2000, Aug 14 2007
Edited by N. J. A. Sloane, Feb 26 2008
Example for n=2 corrected by Adrian Falcone (falcone(AT)gmail.com), Jan 28 2009
Zeroth term inserted by Franklin T. Adams-Watters, Jul 03 2009
a(25)-a(26) from Max Alekseyev, Sep 19 2009
a(27)-a(60) from Max Alekseyev, Sep 07 2016

A054923 Triangle read by rows: number of connected graphs with k >= 0 edges and n nodes (1<=n<=k+1).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 3, 0, 0, 0, 1, 5, 6, 0, 0, 0, 1, 5, 13, 11, 0, 0, 0, 0, 4, 19, 33, 23, 0, 0, 0, 0, 2, 22, 67, 89, 47, 0, 0, 0, 0, 1, 20, 107, 236, 240, 106, 0, 0, 0, 0, 1, 14, 132, 486, 797, 657, 235, 0, 0, 0, 0, 0, 9, 138, 814, 2075, 2678, 1806, 551, 0, 0, 0, 0, 0, 5, 126, 1169, 4495, 8548, 8833, 5026, 1301

Offset: 0

N. J. A. Sloane

The diagonal n = k+1 is A000055(n). - Jonathan Vos Post, Aug 10 2008

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 2;
  0, 0, 0, 2, 3;
  0, 0, 0, 1, 5   6;
  0, 0, 0, 1, 5, 13,  11;
  0, 0, 0, 0, 4, 19,  33,  23;
  0, 0, 0, 0, 2, 22,  67,  89,  47;
  0, 0, 0, 0, 1, 20, 107, 236, 240, 106;
  ... (so with 5 edges there's 1 graph with 4 nodes, 5 with 5 nodes and 6 with 6 nodes). [Typo corrected by Anders Haglund, Jul 08 2008]

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 93, Table 4.2.2; p. 241, Table A2.

Andrew Howroyd, Table of n, a(n) for n = 0..1325
G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017), Table 57.
Gordon Royle, Small graphs
Gus Wiseman, Non-isomorphic representatives of the 12 connected graphs counted in row 5.

Main diagonal is A000055.
Subsequent diagonals give the number of connected unlabeled graphs with n nodes and n+k edges for k=0..2: A001429, A001435, A001436.
Cf. A002905 (row sums), A001349 (column sums), A008406, A046751 (transpose), A054924 (transpose), A046742 (w/o left column), A343088 (labeled).
Cf. A000664, A007718, A050535, A054923, A191646, A191970, A317672, A322114, A322151.

InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i), 1))}
permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p,i->1+x^i)); s/n!}
T(n)={Mat([Col(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])}
{my(A=T(10)); for(n=1, #A, print(A[n,1..n]))} \\ Andrew Howroyd, Oct 23 2019

a(83)-a(89) corrected by Andrew Howroyd, Oct 24 2019

A054924 Triangle read by rows: T(n,k) = number of nonisomorphic unlabeled connected graphs with n nodes and k edges (n >= 1, 0 <= k <= n(n-1)/2).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 1, 1, 0, 0, 0, 0, 3, 5, 5, 4, 2, 1, 1, 0, 0, 0, 0, 0, 6, 13, 19, 22, 20, 14, 9, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 11, 33, 67, 107, 132, 138, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 23, 89, 236, 486, 814, 1169, 1454, 1579, 1515, 1290, 970, 658, 400, 220, 114

Offset: 1

N. J. A. Sloane

			Triangle begins:
1;
0,1;
0,0,1,1;
0,0,0,2,2,1,1;
0,0,0,0,3,5,5,4,2,1,1;
0,0,0,0,0,6,13,19,22,20,14,9,5,2,1,1;
the last batch giving the numbers of connected graphs with 6 nodes and from 0 to 15 edges.

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

R. W. Robinson, Rows 1 to 20 of triangle, flattened (corrected by Sean A. Irvine, Apr 29 2022)
G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.
Sean A. Irvine, Java code (github)
Gordon Royle, Small graphs
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967

Cf. A008406, A054925.
Other versions of this triangle: A046751, A076263, A054923, A046742.
Row sums give A001349, column sums give A002905. A046751 is essentially the same triangle. A054923 and A046742 give same triangle but read by columns.
Main diagonal is A000055. Next diagonal is A001429. Largest entry in each row gives A001437.

A076263 gives a Mathematica program which produces the nonzero entries in each row.
Needs["Combinatorica`"]; Table[Print[row = Join[Array[0&, n-1], Table[ Count[ Combinatorica`ListGraphs[n, k], g_ /; Combinatorica`ConnectedQ[g]], {k, n-1, n*(n-1)/2}]]]; row, {n, 1, 8}] // Flatten (* Jean-François Alcover, Jan 15 2015 *)

A191970 Number of connected graphs with n edges with loops allowed.

Original entry on oeis.org

1, 2, 2, 6, 12, 33, 93, 287, 940, 3309, 12183, 47133, 190061, 796405, 3456405, 15501183, 71681170, 341209173, 1669411182, 8384579797, 43180474608, 227797465130, 1229915324579, 6790642656907, 38311482445514, 220712337683628, 1297542216770482, 7779452884747298

Offset: 0

Alberto Tacchella, Jun 20 2011

nonn

Inverse Euler transform of A053419.
From R. J. Mathar, Jul 25 2017: (Start)
The Multiset Transform gives the number of graphs with n edges (loops allowed) and k components (0<=k<=n):
 1
2
2 3
6 4 4
12 15 6 5
33 36 24 8 6
93 111 64 33 10 7
287 324 207 92 42 12 8
940 1036 633 308 120 51 14 9
3309 3408 2084 966 409 148 60 16 10
12183 11897 6959 3243 1305 510 176 69 18 11
47133 43137 24415 10970 4432 1644 611 204 78 20 12
190061 163608 88402 38763 15125 5628 1983 712 232 87 22 13
796405 644905 332979 140671 53732 19316 6824 2322 813 260 96 24 14
3456405 2639871 1299054 529179 195517 68878 23515 8020 2661 914 288 105 26 15 (End)

			a(1)=2: Either one node with the edge equal to a loop, or two nodes connected by the edge. a(2)=2: Either three nodes on a chain connected by the two edges, or two nodes connected by an edge, one node with a loop. Apparently multi-loops are not allowed (?). - _R. J. Mathar_, Jul 25 2017

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 connected graphs with loops.

Row sums of A322114.

Cf. A000664, A002905, A007718, A050535, A053419, A054923, A191646, A191970, A275421, A322133, A322151, A322152.

\\ See A322114 for InvEulerMT, G.
seq(n)={vecsum([Vec(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])} \\ Andrew Howroyd, Oct 22 2019

Terms a(25) and beyond from Andrew Howroyd, Oct 22 2019

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

A275421 Triangle read by rows: T(n,k) = number of graphs with n edges and k connected components.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 12, 8, 4, 1, 1, 30, 23, 9, 4, 1, 1, 79, 57, 26, 9, 4, 1, 1, 227, 160, 68, 27, 9, 4, 1, 1, 710, 456, 197, 71, 27, 9, 4, 1, 1, 2322, 1402, 567, 208, 72, 27, 9, 4, 1, 1, 8071, 4468, 1748, 604, 211, 72, 27, 9, 4, 1, 1, 29503, 15071, 5555, 1874

Offset: 1

R. J. Mathar, Jul 27 2016

Multiset transformation of A002905.

			      1
      1     1
      3     1     1
      5     4     1     1
     12     8     4     1     1
     30    23     9     4     1     1
     79    57    26     9     4     1     1
    227   160    68    27     9     4     1     1
    710   456   197    71    27     9     4     1     1
   2322  1402   567   208    72    27     9     4     1     1
   8071  4468  1748   604   211    72    27     9     4     1     1
  29503 15071  5555  1874   615   212    72    27     9     4     1

Alois P. Heinz, Rows n = 1..60, flattened
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Table 1.1a, Part 1 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Index entries for triangles generated by the Multiset Transformation

Cf. A002905 (column 1), A000664 (row sums).

rows = 12;
A002905 = Import["https://oeis.org/A002905/b002905.txt", "Table"][[All, 2]];
gf = Product[(1 - y x^j)^-A002905[[j+1]], {j, 1, rows}];
Rest[CoefficientList[#, y]]& /@ Rest[CoefficientList[gf + O[x]^(rows+1), x]] // Flatten (* Jean-François Alcover, May 09 2019, after Alois P. Heinz *)

T(n,1) = A002905(n).

T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1

G.f.: Product_{j>=1} (1-y*x^j)^(-A002905(j)). - Alois P. Heinz, Apr 13 2017

A007719 Number of independent polynomial invariants of symmetric matrix of order n.

Original entry on oeis.org

1, 2, 4, 11, 30, 95, 328, 1211, 4779, 19902, 86682, 393072, 1847264, 8965027, 44814034, 230232789, 1213534723, 6552995689, 36207886517, 204499421849, 1179555353219, 6942908667578, 41673453738272, 254918441681030, 1588256152307002, 10073760672179505

Offset: 0

Colin Mallows

Also, number of connected multigraphs with n edges (allowing loops) and any number of nodes.

Also the number of non-isomorphic connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(3) = 11 connected multiset partitions of {1, 1, 2, 2, 3, 3}:
  (112233),
  (1)(12233), (12)(1233), (112)(233), (123)(123),
  (1)(2)(1233), (1)(12)(233), (1)(23)(123), (12)(13)(23),
  (1)(2)(3)(123), (1)(2)(13)(23).
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..50
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) Table 63.

Row sums of A322115.

Cf. A002905, A007716, A007717, A007719, A020555, A050535, A053419, A076864, A191970, A316972, A316974.

mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++,
  c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {};
  For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t k; s += t]; s!/m];
Kq[q_, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[ Exp[Sum[Kq[q, t, k]/t x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
A007717 = Table[Print[n]; RowSumMats[n, 2 n, 2], {n, 0, 20}];
Join[{1}, EULERi[Rest[A007717]]] (* Jean-François Alcover, Oct 29 2018, using Andrew Howroyd's code for A007717 *)

Inverse Euler transform of A007717.

a(0)=1 added by Alberto Tacchella, Jun 20 2011

a(7)-a(25) from Franklin T. Adams-Watters, Jun 21 2011

A046091 Number of connected planar graphs with n edges.

Original entry on oeis.org

1, 1, 1, 3, 5, 12, 30, 79, 227, 709, 2318, 8049, 29372, 112000, 444855, 1833072, 7806724, 34252145, 154342391, 712231465, 3357126655, 16119421175, 78580665333

Offset: 0

Brendan McKay

Inverse Euler transform of A343872. - Andrew Howroyd, May 05 2021

			a(3) = 3 since the three connected graphs with three edges are a path, a triangle and a "Y".
The first difference between this sequence and A002905 is for n=9 edges where we see K_{3,3}, the "utility graph".

B. D. McKay and A. Piperno, Practical Graph Isomorphism, II, J. Symbolic Computation, 60 (2014), pp. 94-112.
Eric Weisstein's World of Mathematics, Planar Connected Graph.

Row sums of A343873.
Column sums of A049334.
Cf. A002905, A003094, A066951, A291842, A343869, A343872.

# count graphs for the sequence by number of vertices v, sum over v afterwards
geng -c $v $n:$n | planarg -q | countg -q # Georg Grasegger, Jul 06 2023

a(11)-a(19) from Martin Fuller using nauty by Brendan McKay, Mar 07 2015

a(20)-a(22) added by Georg Grasegger, Jul 06 2023

A322137 Number of labeled connected graphs with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 1, 3, 17, 140, 1524, 20673, 336259, 6382302, 138525780, 3384988809, 91976158434, 2751122721402, 89833276321440, 3179852538140115, 121287919647418118, 4959343701136929850, 216406753768138678671, 10037782414506891597734, 493175891246093032826160

Offset: 0

Gus Wiseman, Nov 27 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
P. S. Kolesnikov and B. K. Sartayev, On the special identities of Gelfand--Dorfman algebras, arXiv:2105.13815 [math.RA], 2021.
Gus Wiseman, The a(4) = 140 labeled connected graphs with 4 edges.

Cf. A000664, A002905, A007718, A013922, A054923, A057500, A191646, A275421, A291842 (planar case), A322114, A322115.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n+1],{2}],{n}],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,6}]

Connected(v)={my(u=vector(#v));for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1,k)*v[k]*u[n-k])); u}
seq(n)={Vec(vecsum(Connected(vector(2*n, j, (1 + x + O(x*x^n))^binomial(j,2)))))} \\ Andrew Howroyd, Nov 28 2018

Terms a(8) and beyond from Andrew Howroyd, Nov 28 2018

Showing 1-10 of 28 results. Next

cp's OEIS Frontend

A008406 Triangle T(n,k) read by rows, giving number of graphs with n nodes (n >= 1) and k edges (0 <= k <= n(n-1)/2).

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A000664 Number of graphs with n edges.

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A054923 Triangle read by rows: number of connected graphs with k >= 0 edges and n nodes (1<=n<=k+1).

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A054924 Triangle read by rows: T(n,k) = number of nonisomorphic unlabeled connected graphs with n nodes and k edges (n >= 1, 0 <= k <= n(n-1)/2).

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A191970 Number of connected graphs with n edges with loops allowed.

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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.

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A275421 Triangle read by rows: T(n,k) = number of graphs with n edges and k connected components.

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