A004108 -id:A004108 - cp's OEIS frontend

A004110 Number of n-node unlabeled graphs without endpoints (i.e., no nodes of degree 1).

1, 1, 1, 2, 5, 16, 78, 588, 8047, 205914, 10014882, 912908876, 154636289460, 48597794716736, 28412296651708628, 31024938435794151088, 63533059372622888758054, 244916078509480823407040988, 1783406527599529094009748567708, 24605674623474428415849066062642456

Offset: 0

N. J. A. Sloane

nonn

a(n) is also the number of unlabeled mating graphs with n nodes. A mating graph has no two vertices with identical sets of neighbors. - Tanya Khovanova, Oct 23 2008

F. Harary, Graph Theory, Wiley, 1969. See illustrations in Appendix 1.
F. Harary and E. Palmer, Graphical Enumeration, (1973), compare formula (8.7.11).
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..26 from R. W. Robinson)
David Cook II, Nested colourings of graphs, arXiv preprint arXiv:1306.0140 [math.CO], 2013.
Ira M. Gessel and Ji Li, Enumeration of point-determining graphs, J. Combinatorial Theory Ser. A 118 (2011), 591-612.
Hemanshu Kaul and Jeffrey A. Mudrock, Counting List Colorings of Unlabeled Graphs, arXiv:2409.06063 [math.CO], 2024. See p. 6.
R. J. Mathar, Illustrations for n=1..5 nodes.
Ronald C. Read, The enumeration of mating-type graphs, Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.
R. W. Robinson, Graphs without endpoints - computer printout.
N. J. A. Sloane, Illustration of a(0)-a(5).

Row sums of A123551.
Cf. A059166 (n-node connected labeled graphs without endpoints), A059167 (n-node labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A006024 (number of labeled mating graphs with n nodes), A129584 (bi-point-determining graphs).
If isolated nodes are forbidden, see A261919.
Cf. A000088.

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_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
a[n_] := Sum[permcount[p] * 2^edges[p] * Coefficient[Product[1 - x^p[[i]], {i, 1, Length[p]}], x, n - k]/k!, {k, 1, n}, {p, IntegerPartitions[k]}]; a[0] = 1;
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 27 2018, after Andrew Howroyd *)

\\ Compare A000088.
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) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
a(n) = {my(s=0); sum(k=1, n, forpart(p=k, s+=permcount(p) * 2^edges(p) * polcoef(prod(i=1, #p, 1-x^p[i]), n-k)/k!)); s} \\ Andrew Howroyd, Sep 09 2018

A059167 Number of n-node labeled graphs without endpoints.

Original entry on oeis.org

1, 1, 1, 2, 15, 314, 13757, 1142968, 178281041, 52610850316, 29702573255587, 32446427369694348, 69254848513798160815, 291053505824567573585744, 2421830049319361003822380177, 40050220743831370293688592267252, 1319550593412053164173947687592553185

Offset: 0

Vladeta Jovovic, Jan 12 2001

F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 31, problem 1.16(a).

Robert Israel, Table of n, a(n) for n = 0..81
Marko R. Riedel, Geoffrey Critzer, Math.Stackexchange.com, Proof of the closed form of the e.g.f. by combinatorial species. - _Marko Riedel_, Sep 18 2016

Column k=0 of A327369.

Cf. A059166 (n-node connected labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A004110 (n-node unlabeled graphs without endpoints).

F:= proc(N) local S;
   S:= series(exp(1/2*x^2)*Sum(2^binomial(n, 2)*(x/exp(x))^n/n!, n = 0 .. N),x,N+1);
   seq(coeff(S,x,i)*i!,i=0..N)
end proc:
F(20); # Robert Israel, Sep 18 2016

b[n_] := If[n < 3, Boole[n == 1], n!*Sum[(-1)^(n - j) * SeriesCoefficient[1 + Log[Sum[2^(k*(k - 1)/2)*x^k/k!, {k, 0, j}]], {x, 0, j}] * j^(n - j)/(n - j)!, {j, 0, n}]]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, i] b[i + 1] a[n - i - 1], {i, 0, n - 1}]; Table[a@ n, {n, 0, 15}] (* Michael De Vlieger, Sep 19 2016, after Vaclav Kotesovec at A059166 *)

seq(n)={my(A=x/exp(x + O(x^n))); Vec(serlaplace(exp(x^2/2 + O(x*x^n)) * sum(k=0, n, 2^binomial(k, 2)*A^k/k!)))} \\ Andrew Howroyd, Sep 09 2018

a(n) = Sum_{i=0..n-1} binomial(n-1, i)*b(i+1)*a(n-i-1), n>0, a(0)=1, where b(n) is number of n-node connected labeled graphs without endpoints (Cf. A059166).
E.g.f.: exp(x^2/2)*(Sum_{n >= 0} 2^binomial(n, 2)*(x/exp(x))^n/n!). - Vladeta Jovovic, Mar 23 2004
a(n) ~ 2^(n*(n-1)/2). - Vaclav Kotesovec, Sep 22 2016

More terms from John Renze (jrenze(AT)yahoo.com), Feb 01 2001

A059166 Number of n-node connected labeled graphs without endpoints.

Original entry on oeis.org

1, 1, 0, 1, 10, 253, 12058, 1052443, 169488200, 51045018089, 29184193354806, 32122530765469967, 68867427921051098084, 290155706369032525823085, 2417761578629525173499004146, 40013923790443379076988789688611, 1318910080173114018084245406769861936

Offset: 0

Vladeta Jovovic, Jan 12 2001

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 404.

Vaclav Kotesovec, Table of n, a(n) for n = 0..80

Cf. A059167 (n-node labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A004110 (n-node unlabeled graphs without endpoints).

Cf. A001187, A007146, A095983, A100743, A261919, A322395.

c:= 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)*c(k), k=1..n-1)/n)
    end:
a:= n-> max(0, add((-1)^i*binomial(n, i)*c(n-i)*(n-i)^i, i=0..n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 27 2017

Flatten[{1,1,0,Table[n!*Sum[(-1)^(n-j)*SeriesCoefficient[1+Log[Sum[2^(k*(k-1)/2)*x^k/k!,{k,0,j}]],{x,0,j}]*j^(n-j)/(n-j)!,{j,0,n}],{n,3,15}]}] (* Vaclav Kotesovec, May 14 2015 *)
c[0] = 1; c[n_] := c[n] = 2^(n*(n-1)/2) - Sum[k*Binomial[n, k]*2^((n-k)*(n - k - 1)/2)*c[k], {k, 1, n-1}]/n; a[0] = a[1] = 1; a[2] = 0; a[n_] := Sum[(-1)^i*Binomial[n, i]*c[n-i]*(n-i)^i, {i, 0, n}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 27 2017, using Alois P. Heinz's code for c(n) *)

seq(n)={Vec(serlaplace(1 + x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!))))} \\ Andrew Howroyd, Sep 09 2018

a(n) = Sum_{i=0..n} (-1)^i*binomial(n, i)*c(n-i)*(n-i)^i, for n>2, a(0)=1, a(1)=1, a(2)=0, where c(n) is number of n-node connected labeled graphs (cf. A001187).
E.g.f.: 1 + x^2/2 + log(Sum_{n >= 0} 2^binomial(n, 2)*(x*exp(-x))^n/n!).
a(n) ~ 2^(n*(n-1)/2). - Vaclav Kotesovec, May 14 2015
Logarithmic transform of A100743, if we assume a(1) = 0. - Gus Wiseman, Aug 15 2019

More terms from John Renze (jrenze(AT)yahoo.com), Feb 01 2001

A261919 Number of n-node unlabeled graphs without isolated nodes or endpoints (i.e., no nodes of degree 0 or 1).

Original entry on oeis.org

1, 0, 0, 1, 3, 11, 62, 510, 7459, 197867, 9808968, 902893994, 153723380584, 48443158427276, 28363698856991892, 30996526139142442460, 63502034434187094606966, 244852545450108200518282934, 1783161611521019613186341526720, 24603891216946828886755056314074748

Offset: 0

N. J. A. Sloane, Sep 15 2015

nonn

			From _Gus Wiseman_, Aug 15 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(5) = 11 graphs (empty columns not shown):
  {}  {12,13,23}  {12,13,24,34}        {12,13,24,35,45}
                  {13,14,23,24,34}     {12,14,25,34,35,45}
                  {12,13,14,23,24,34}  {12,15,25,34,35,45}
                                       {13,14,23,24,35,45}
                                       {12,13,24,25,34,35,45}
                                       {13,15,24,25,34,35,45}
                                       {14,15,24,25,34,35,45}
                                       {12,13,15,24,25,34,35,45}
                                       {14,15,23,24,25,34,35,45}
                                       {13,14,15,23,24,25,34,35,45}
                                       {12,13,14,15,23,24,25,34,35,45}
(End)

F. Harary, Graph Theory, Wiley, 1969. See illustrations in Appendix 1.

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 1..26 from Max Alekseyev)
N. J. A. Sloane, Illustration of a(0)-a(5) [Ignore the graphs with isolated nodes]

Cf. A004108 (connected version), A004110 (version allowing isolated nodes).
The labeled version is A100743.
Cf. A003227, A006125, A007146, A059166, A095983, A322396.

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_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
b[n_] := Sum[permcount[p]*2^edges[p]*Coefficient[Product[1-x^p[[i]], {i, 1, Length[p]}], x, n-k]/k!, {k, 1, n}, {p, IntegerPartitions[k]}]; b[0] = 1;
a[n_] := b[n] - b[n-1];
a /@ Range[0, 19] (* Jean-François Alcover, Sep 12 2019, after Andrew Howroyd in A004110 *)

First differences of A004110: a(n) = A004110(n)-A004110(n-1).

Euler transform of A004108, if we assume A004108(1) = 0. - Gus Wiseman, Aug 15 2019

a(1)-a(11) computed by Brendan McKay, Sep 15 2015
a(12)-a(26) computed from A004110 by Max Alekseyev, Sep 16 2015
a(0) = 1 prepended by Gus Wiseman, Aug 15 2019

A092430 Number of n-node labeled connected mating graphs, cf. A006024.

Original entry on oeis.org

1, 1, 25, 438, 18388, 1409674, 206682994, 58152537184, 31715884061624, 33827568738189576, 71066571962396085656, 295645506683051376527648, 2444503529745123474354656720, 40269655263141217619453414445968

Offset: 2

Goran Kilibarda, Vladeta Jovovic, Mar 22 2004

nonn

Number of n-node unlabeled connected mating graphs = number of n-node unlabeled connected graphs without endpoints, n>2; cf. A004108.
The number of graphs of this type with n>=1 nodes and 1<=k<=n components defines the triangle
0;
1,0;
1,0,0;
25,3,0,0;
438,10,0,0,0;
18388,385,15,0,0,0;
1409674,10073,105,0,0,0,0;
206682994,561267,5530,105,0,0,0,0;
58152537184,53672556,197344,1260,0,0,0,0,0;
with row sums A007833. - R. J. Mathar, Apr 29 2019

Goran Kilibarda, "Enumeration of unlabeled mating graphs", Belgrade, 2004, to be published.

Andrew Howroyd, Table of n, a(n) for n = 2..50
Goran Kilibarda, Enumeration of Unlabeled Mating Graphs, Graphs and Combinatorics, Volume 23, Number 2 / April, 2007, pp. 183-199.

Cf. A006024, A079306, A007833, A004108.

Cf. A059166.

a(n)={n!*polcoef(log(sum(i=0, n, 2^binomial(i, 2)*log(1+x + O(x*x^n))^i/i!)/(1+x)), n)} \\ Andrew Howroyd, Sep 09 2018

From Vladeta Jovovic, Mar 28 2004: (Start)
E.g.f.: log((Sum_{n>=0} 2^binomial(n, 2)*log(1+x)^n/n!)/(1+x)).
a(n) = A079306(n) + (-1)^n*(n-1)!. (End)

A129584 Number of unlabeled bi-point-determining graphs: graphs in which no two vertices have the same neighborhoods or the same augmented neighborhoods (the augmented neighborhood of a vertex is the neighborhood of the vertex union the vertex itself).

Original entry on oeis.org

1, 0, 0, 1, 6, 36, 324, 5280, 156088, 8415760

Offset: 1

Ji Li (vieplivee(AT)hotmail.com), May 07 2007

This is the unlabeled case of bi-point-determining graphs, which are basically graphs that are both point-determining (no two vertices have the same neighborhoods) and co-point-determining (graphs whose complements are point-determining)

Ira M. Gessel and Ji Li, Enumeration of point-determining Graphs, Journal of Combinatorial Theory, Series A 118 (2011) 591-612.

Cf. graphs: labeled A006125, unlabeled A000568; connected graphs: labeled A001187, unlabeled A001349; point-determining graphs: labeled A006024, unlabeled A004110; connected point-determining graphs: labeled A092430, unlabeled A004108; connected co-point-determining graphs: labeled A079306, unlabeled A004108; bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.

A342556 a(n) is the number of unlabeled connected graphs without endpoints with n edges.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 5, 9, 22, 60, 164, 494, 1624, 5602, 20600, 79813, 323806, 1371025, 6034105, 27513424, 129641411, 629862824, 3149541908, 16183100922, 85328280263, 461123500894, 2551342936264, 14438734591483, 83506198920054, 493163726073210, 2971858162771887

Offset: 0

Hugo Pfoertner, May 21 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50
Brendan McKay and Adolfo Piperno, Practical Graph Isomorphism, II, Journal of Symbolic Computation, 60 (2014), pp. 94-112.
Brendan McKay and Adolfo Piperno, nauty and Traces, programs for computing automorphism groups of graphs and digraphs.

Row sums of A342557.

Cf. A004108 (n vertices), A307317 (multigraph), A369290 (not necessarily connected).

Inverse Euler transform of A369290.

a(0)=1 prepended and a(25) onwards from Andrew Howroyd, Feb 01 2024

A342557 T(n,m) is the number of unlabeled connected graphs without endpoints on m nodes with n edges, where T(n,m), m <= n, is a triangle read by rows.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 3, 5, 1, 0, 0, 0, 0, 2, 11, 8, 1, 0, 0, 0, 0, 1, 15, 31, 12, 1, 0, 0, 0, 0, 1, 12, 63, 71, 16, 1, 0, 0, 0, 0, 0, 8, 89, 231, 144, 21, 1, 0, 0, 0, 0, 0, 5, 97, 513, 707, 274, 27, 1

Offset: 1

Hugo Pfoertner, May 21 2021

The number of nonzero terms in the n-th row is A083920(n-1). - Hugo Pfoertner, Feb 01 2024

			The triangle begins
  0;
  0, 0;
  0, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 1, 3,  1;
  0, 0, 0, 0, 3,  5,  1;
  0, 0, 0, 0, 2, 11,  8,  1;
  0, 0, 0, 0, 1, 15, 31, 12,  1;
  0, 0, 0, 0, 1, 12, 63, 71, 16, 1;

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50; first 24 rows from Hugo Pfoertner)

Cf. A004108 (column sums), A342556 (row sums).

Cf. A083920, A369932 (not necessarily connected).

\\ Needs G() defined in A369932.
InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
T(n)={my(r=Vec(InvEulerMTS(substvec(G(n),[x,y],[y,x])))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y),i)) }
{ my(A=T(12)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 07 2024

Bivariate inverse Euler transform of A369932. - Andrew Howroyd, Feb 07 2024

A129583 Number of labeled bi-point-determining graphs with n vertices.

Original entry on oeis.org

1, 1, 0, 0, 12, 312, 13824, 1147488, 178672128, 52666091712, 29715982846848, 32452221242518272, 69259424722321036032, 291060255757818125657088, 2421848956937579216663491584, 40050322614433939228627991906304, 1319551659023608317386779165849208832

Offset: 0

Ji Li (vieplivee(AT)hotmail.com), May 07 2007

A bi-point determining graph is a graph in which no two vertices have the same neighborhoods or the same augmented neighborhoods (the augmented neighborhood of a vertex is the neighborhood of the vertex union the vertex itself).

R. C. Read, The Enumeration of Mating-Type Graphs. Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Ira M. Gessel and Ji Li, Enumeration of point-determining Graphs, Journal of Combinatorial Theory, Series A 118 (2011) 591-612.

Cf. graphs: labeled A006125, unlabeled A000568; connected graphs: labeled A001187, unlabeled A001349; point-determining graphs: labeled A006024, unlabeled A004110; connected point-determining graphs: labeled A092430, unlabeled A004108; connected co-point-determining graphs: labeled A079306, unlabeled A004108; bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.

seq(n)={my(g=sum(k=0, n, 2^binomial(k,2)*x^k/k!) + O(x*x^n)); Vec(serlaplace(subst(g, x, 2*log(1+x+O(x*x^n))-x)))} \\ Andrew Howroyd, May 06 2021

E.g.f.: G(2*log(1+x)-x) where G(x) is the e.g.f. of A006125.

a(0)=1 prepended and terms a(13) and beyond from Andrew Howroyd, May 06 2021

A129585 Number of labeled connected bi-point-determining graphs with n vertices (see A129583).

Original entry on oeis.org

1, 1, 0, 0, 12, 252, 12312, 1061304, 170176656, 51134075424, 29204599254624, 32130964585236096, 68873851786953047040, 290164895151435531345024, 2417786648013402212500060416, 40014055814155246577685250570752, 1318911434129029730677931158374449664

Offset: 0

Ji Li (vieplivee(AT)hotmail.com), May 07 2007

The calculation of connected bi-point-determining graphs is carried out by examining the connected components of bi-point-determining graphs. For more details, see reference.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Ira M. Gessel and Ji Li, Enumeration of point-determining Graphs, Journal of Combinatorial Theory, Series A 118 (2011) 591-612.

Cf. graphs: labeled A006125, unlabeled A000568; connected graphs: labeled A001187, unlabeled A001349; point-determining graphs: labeled A006024, unlabeled A004110; connected point-determining graphs: labeled A092430, unlabeled A004108; connected co-point-determining graphs: labeled A079306, unlabeled A004108; bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.

max = 15; f[x_] := x + Log[ Sum[ 2^Binomial[n, 2]*((2*Log[1 + x] - x)^n/n!), {n, 0, max}]/(1 + x)]; A129585 = Drop[ CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]!, 1](* Jean-François Alcover, Jan 13 2012, after e.g.f. *)

seq(n)={my(g=sum(k=0, n, 2^binomial(k,2)*x^k/k!) + O(x*x^n)); Vec(serlaplace(1+x+log(subst(g, x, 2*log(1+x+O(x*x^n))-x)/(1+x))))} \\ Andrew Howroyd, May 06 2021

E.g.f.: 1 + x + log((Sum_{n>=0} 2^binomial(n,2)*(2*log(1+x)-x)^n/n!)/(1+x)). - Goran Kilibarda, Vladeta Jovovic, May 09 2007

E.g.f.: 1 + x + log(B(x)/(1+x)) where B(x) is the e.g.f. of A129583. - Andrew Howroyd, May 06 2021

More terms from Goran Kilibarda, Vladeta Jovovic, May 09 2007

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, May 06 2021

cp's OEIS Frontend

A004110 Number of n-node unlabeled graphs without endpoints (i.e., no nodes of degree 1).

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A059167 Number of n-node labeled graphs without endpoints.

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A059166 Number of n-node connected labeled graphs without endpoints.

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Maple

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A261919 Number of n-node unlabeled graphs without isolated nodes or endpoints (i.e., no nodes of degree 0 or 1).

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A092430 Number of n-node labeled connected mating graphs, cf. A006024.

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A129584 Number of unlabeled bi-point-determining graphs: graphs in which no two vertices have the same neighborhoods or the same augmented neighborhoods (the augmented neighborhood of a vertex is the neighborhood of the vertex union the vertex itself).

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A342556 a(n) is the number of unlabeled connected graphs without endpoints with n edges.

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A342557 T(n,m) is the number of unlabeled connected graphs without endpoints on m nodes with n edges, where T(n,m), m <= n, is a triangle read by rows.

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