A261919 -id:A261919 - 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

A007146 Number of unlabeled simple connected bridgeless graphs with n nodes.

Original entry on oeis.org

1, 0, 1, 3, 11, 60, 502, 7403, 197442, 9804368, 902818087, 153721215608, 48443044675155, 28363687700395422, 30996524108446916915, 63502033750022111383196, 244852545022627009655180986, 1783161611023802810566806448531, 24603891215865809635944516464394339

Offset: 1

N. J. A. Sloane

Also unlabeled simple graphs with spanning edge-connectivity >= 2. The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices. - Gus Wiseman, Sep 02 2019

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 = 1..40 (terms 1..22 from R. J. Mathar)
P. Hanlon and R. W. Robinson, Counting bridgeless graphs, J. Combin. Theory, B 33 (1982), 276-305, Table III.
Eric Weisstein's World of Mathematics, Bridgeless Graph
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Simple Graph
Gus Wiseman, The a(3) = 1 through a(5) = 11 connected bridgeless graphs.

Cf. A005470 (number of simple graphs).
Cf. A007145 (number of simple connected rooted bridgeless graphs).
Cf. A052446 (number of simple connected bridged graphs).
Cf. A263914 (number of simple bridgeless graphs).
Cf. A263915 (number of simple bridged graphs).
The labeled version is A095983.
Row sums of A263296 if the first two columns are removed.
BII-numbers of set-systems with spanning edge-connectivity >= 2 are A327109.
Graphs with non-spanning edge-connectivity >= 2 are A327200.
2-vertex-connected graphs are A013922.
Cf. A000719, A001349, A002494, A261919, A327069, A327071, A327074, A327075, A327077, A327109, A327144, A327146.

\\ Translation of theorem 3.2 in Hanlon and Robinson reference. See A004115 for graphsSeries and A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(gc=sLog(graphsSeries(n)), gcr=sPoint(gc)); sSolve( gc + gcr^2/2 - sRaise(gcr,2)/2, x*sv(1)*sExp(gcr) )}
NumUnlabeledObjsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 31 2020

a(n) = A001349(n) - A052446(n). - Gus Wiseman, Sep 02 2019

Reference gives first 22 terms.

A004108 Number of n-node unlabeled connected graphs without endpoints.

Original entry on oeis.org

1, 1, 0, 1, 3, 11, 61, 507, 7442, 197772, 9808209, 902884343, 153723152913, 48443147912137, 28363697921914475, 30996525982586676021, 63502034385272108655525, 244852545421597419740767106, 1783161611489937453151313949442, 24603891216883233547700609793901996

Offset: 0

N. J. A. Sloane

nonn

Also number of n-node unlabeled connected mating graphs, cf. A006024 and A092430 (conjectured by Vladeta Jovovic, proved by G. Kilibarda). - Vladeta Jovovic, Oct 07 2004

F. Harary and E. Palmer, Graphical Enumeration, (1973), formula (8.7.11).
Goran Kilibarda, "Enumeration of unlabeled mating graphs", Belgrade, 2004, to be published.
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
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 1..26 from R. W. Robinson)
David Cook II, Nested colourings of graphs, arXiv preprint arXiv:1306.0140 [math.CO], 2013.
Gabe Cunningham and Igor Minevich, Graph Products of Groups, arXiv:2502.05648 [math.GR], 2025. See p. 5.
Hemanshu Kaul and Jeffrey A. Mudrock, Counting List Colorings of Unlabeled Graphs, arXiv:2409.06063 [math.CO], 2024. See p. 6.
Goran Kilibarda, Enumeration of Unlabeled Mating Graphs, Graphs and Combinatorics, Volume 23, Number 2 / April, 2007, pp. 183-199.
R. W. Robinson, Connected graphs without endpoints - computer printout.
Eric Weisstein's World of Mathematics, Connected Graph.

Cf. A059166 (n-node connected labeled graphs without endpoints), A059167 (n-node labeled graphs without endpoints), A004110 (Euler Transform, n-node unlabeled graphs without endpoints).
Cf. A006024, A092430 (n-node labeled connected mating graphs).
See also A261919.
Cf. A342556, A342557.
Counts include those for distance-critical graphs, A349402.

terms = 19;
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];
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]}];
A004110 = Table[b[n], {n, 1, terms-1}];
Join[{1}, EULERi[A004110]] (* Jean-François Alcover, Jan 21 2019, after Andrew Howroyd *)

Inverse Euler transform of A004110. - Andrew Howroyd, Sep 09 2018

a(0)=1 prepended by Andrew Howroyd, Sep 09 2018

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

A100743 Number of labeled n-vertex graphs without vertices of degree <=1.

Original entry on oeis.org

1, 0, 0, 1, 10, 253, 12068, 1052793, 169505868, 51046350021, 29184353055900, 32122563615242615, 68867440268165982320, 290155715157676330952559, 2417761590648159731258579164, 40013923822242935823157820555477, 1318910080336893719646370269435043184

Offset: 0

Goran Kilibarda, Zoran Maksimovic, Vladeta Jovovic, Jan 03 2005

nonn

			From _Gus Wiseman_, Aug 18 2019: (Start)
The a(4) = 10 edge-sets:
  {12,13,24,34}
  {12,14,23,34}
  {13,14,23,24}
  {12,13,14,23,24}
  {12,13,14,23,34}
  {12,13,14,24,34}
  {12,13,23,24,34}
  {12,14,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}
(End)

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

Graphs without isolated nodes are A006129.
The connected case is A059166.
Graphs without endpoints are A059167.
Labeled graphs with endpoints are A245797.
The unlabeled version is A261919.
Cf. A095983, A136284, A322395, A327079, A327107, A327227, A327229, A327230.
Cf. A095983, A322395.

m = 13;
egf = Exp[-x + x^2/2]*Sum[2^(n (n-1)/2)*(x/Exp[x])^n/n!, {n, 0, m+1}];
s = egf + O[x]^(m+1);
a[n_] := n!*SeriesCoefficient[s, n];
Table[a[n], {n, 0, m}] (* Jean-François Alcover, Feb 23 2019 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}] (* Gus Wiseman, Aug 18 2019 *)

seq(n)={Vec(serlaplace(exp(-x + x^2/2 + O(x*x^n))*sum(k=0, n, 2^(k*(k-1)/2)*(x/exp(x + O(x^n)))^k/k!)))} \\ Andrew Howroyd, Sep 04 2019

E.g.f.: exp(-x+x^2/2)*(Sum_{n>=0} 2^(n*(n-1)/2)*(x/exp(x))^n/n!). - Vladeta Jovovic, Jan 26 2006
Exponential transform of A059166. - Gus Wiseman, Aug 18 2019
Inverse binomial transform of A059167. - Gus Wiseman, Sep 02 2019

Terms a(14) and beyond from Andrew Howroyd, Sep 04 2019

A327230 Number of non-isomorphic set-systems covering n vertices with at least one endpoint/leaf.

Original entry on oeis.org

0, 1, 3, 14, 198

Offset: 0

Gus Wiseman, Sep 01 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.

Also covering set-systems with minimum vertex-degree 1.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 14 set-systems:
  {{1}}  {{1,2}}      {{1,2,3}}
         {{1},{2}}    {{1},{2,3}}
         {{2},{1,2}}  {{1},{2},{3}}
                      {{1,3},{2,3}}
                      {{3},{1,2,3}}
                      {{1},{3},{2,3}}
                      {{2,3},{1,2,3}}
                      {{2},{1,3},{2,3}}
                      {{2},{3},{1,2,3}}
                      {{3},{1,3},{2,3}}
                      {{1},{2},{3},{2,3}}
                      {{3},{2,3},{1,2,3}}
                      {{2},{3},{1,3},{2,3}}
                      {{2},{3},{2,3},{1,2,3}}

Unlabeled covering set-systems are A055621.
The labeled version is A327229.
The non-covering version is A327335 (partial sums).
Cf. A002494, A245797, A261919, A283877, A327103, A327105, A327197, A327227, A327228.

A294217 Triangle read by rows: T(n,k) is the number of graphs with n vertices and minimum vertex degree k, (0 <= k < n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 4, 2, 1, 11, 12, 8, 2, 1, 34, 60, 43, 15, 3, 1, 156, 378, 360, 121, 25, 3, 1, 1044, 3843, 4869, 2166, 378, 41, 4, 1, 12346, 64455, 113622, 68774, 14306, 1095, 65, 4, 1, 274668, 1921532, 4605833, 3953162, 1141597, 104829, 3441, 100, 5, 1

Offset: 1

Eric W. Weisstein, Oct 25 2017

Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A327366. Burnside's lemma can be used to extend this method to the unlabeled case. - Andrew Howroyd, Mar 10 2020

			Triangle begins:
    1;
    1,   1;
    2,   1,   1;
    4,   4,   2,   1;
   11,  12,   8,   2,  1;
   34,  60,  43,  15,  3, 1;
  156, 378, 360, 121, 25, 3, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..210 (first 20 rows)
Eric Weisstein's World of Mathematics, Minimum Vertex Degree

Row sums are A000088 (simple graphs on n nodes).
Columns k=0..2 are A000088(n-1), A324693, A324670.
Cf. A263293 (triangle of n-node maximum vertex degree counts).
The labeled version is A327366.
Cf. A002494, A004110, A261919, A327227, A327230, A327335, A327372.

T(n, 0) = A000088(n-1).
T(n, n-2) = A004526(n) for n > 1.
T(n, n-1) = 1.
T(n, k) = A263293(n, n-1-k). - Andrew Howroyd, Sep 03 2019

A327335 Number of non-isomorphic set-systems with n vertices and at least one endpoint/leaf.

Original entry on oeis.org

0, 1, 4, 18, 216

Offset: 0

Gus Wiseman, Sep 02 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.

Also covering set-systems with minimum covered vertex-degree 1.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 18 set-systems:
  {{1}}  {{1}}        {{1}}
         {{1,2}}      {{1,2}}
         {{1},{2}}    {{1},{2}}
         {{1},{1,2}}  {{1,2,3}}
                      {{1},{1,2}}
                      {{1},{2,3}}
                      {{1},{1,2,3}}
                      {{1,2},{1,3}}
                      {{1},{2},{3}}
                      {{1,2},{1,2,3}}
                      {{1},{2},{1,3}}
                      {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{2},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1},{2},{3},{1,2}}
                      {{1},{2},{1,2},{1,3}}
                      {{1},{2},{1,2},{1,2,3}}

Unlabeled set-systems are A000612.
The labeled version is A327228.
The covering version is A327230 (first differences).
Cf. A002494, A245797, A261919, A283877, A327103, A327105, A327197, A327227, A327229, A327336.

A327372 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 3, 1, 1, 1, 1, 11, 5, 4, 1, 2, 0, 62, 29, 18, 6, 4, 2, 1, 510, 225, 101, 32, 13, 4, 3, 0, 7459, 2674, 842, 223, 72, 19, 9, 3, 1, 197867, 50834, 10784, 2171, 504, 115, 34, 9, 4, 0, 9808968, 1653859, 228863, 32322, 5268, 944, 209, 46, 16, 4, 1

Offset: 0

Gus Wiseman, Sep 04 2019

			Triangle begins:
   1
   0  0
   0  0  1
   1  0  1  0
   3  1  1  1  1
  11  5  4  1  2  0

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Gus Wiseman, The unlabeled graphs covering 5 vertices with k endpoints.

Row sums are A002494.
Column k = 0 is A261919.
The non-covering version is A327371.
The labeled version is A327377.
Cf. A000088, A004110, A294217, A327227, A327369, A327370.

\\ Needs G(n) defined in A327371.
T(n)={my(v=Vec(G(n)*(1 - x))); vector(#v, n, Vecrev(v[n], n))}
my(A=T(10)); for(n=1, #A, print(A[n])) \\ Andrew Howroyd, Jan 11 2024

Column-wise first differences of A327371.

Terms a(21) and beyond from Andrew Howroyd, Sep 11 2019

A369932 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs with n edges and k vertices and without endpoints or isolated vertices.

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, 2, 0, 0, 0, 0, 3, 5, 2, 0, 0, 0, 0, 2, 11, 9, 3, 0, 0, 0, 0, 1, 15, 32, 16, 4, 0, 0, 0, 0, 1, 12, 63, 76, 25, 5, 0, 0, 0, 0, 0, 8, 89, 234, 162, 39, 6, 0, 0, 0, 0, 0, 5, 97, 515, 730, 332, 60, 9

Offset: 1

Andrew Howroyd, Feb 07 2024

			Triangle begins:
  0;
  0, 0;
  0, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 1, 3,  2;
  0, 0, 0, 0, 3,  5,  2;
  0, 0, 0, 0, 2, 11,  9,  3;
  0, 0, 0, 0, 1, 15, 32, 16,  4;
  0, 0, 0, 0, 1, 12, 63, 76, 25, 5;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A369290.
Column sums are A261919.
Main diagonal is A008483.
Cf. A342557 (connected), A123551 (without endpoints).

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) = {my(s=O(x*x^n)); sum(k=0, n, forpart(p=k, s+=permcount(p) * edges(p, w->1+y^w+O(y*y^n)) * x^k * prod(i=1, #p, 1-(y*x)^p[i], 1+O(x^(n-k+1))) / k!)); s*(1-x)}
T(n)={my(r=Vec(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])) }

T(n,k) = A123551(k,n) - A123551(k-1,n).

cp's OEIS Frontend

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

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A007146 Number of unlabeled simple connected bridgeless graphs with n nodes.

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A004108 Number of n-node unlabeled connected graphs without endpoints.

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

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A100743 Number of labeled n-vertex graphs without vertices of degree <=1.

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A327230 Number of non-isomorphic set-systems covering n vertices with at least one endpoint/leaf.

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A294217 Triangle read by rows: T(n,k) is the number of graphs with n vertices and minimum vertex degree k, (0 <= k < n).

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A327335 Number of non-isomorphic set-systems with n vertices and at least one endpoint/leaf.

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