A339645 -id:A339645 - cp's OEIS frontend

A002218 Number of unlabeled nonseparable (or 2-connected) graphs (or blocks) with n nodes.

0, 1, 1, 3, 10, 56, 468, 7123, 194066, 9743542, 900969091, 153620333545, 48432939150704, 28361824488394169, 30995890806033380784, 63501635429109597504951, 244852079292073376010411280, 1783160594069429925952824734641, 24603887051350945867492816663958981

Offset: 1

N. J. A. Sloane

By definition, a(n) gives the number of graphs with zero cutpoints. - Travis Hoppe, Apr 28 2014
For n > 2, a(n) is also the number of simple biconnected graphs on n nodes. - Eric W. Weisstein, Dec 07 2021
This sequence follows R. W. Robinson's definition of a nonseparable graph which includes K_2 but not the singleton graph K_1. This definition is most suited to graphical enumeration. Other authors sometimes include K_1 as a block or exclude K_2 as not 2-connected. - Andrew Howroyd, Feb 26 2023

P. Butler and R. W. Robinson, On the computer calculation of the number of nonseparable graphs, pp. 191 - 208 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 188.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.
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).

Andrew Howroyd, Table of n, a(n) for n = 1..40 [terms 1..26 from R. W. Robinson]
P. Butler and R. W. Robinson, On the computer calculation of the number of nonseparable graphs, pp. 191 - 208 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp. [Annotated scanned copy]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
R. W. Robinson, Computer print-out of first 26 terms [Annotated scanned copy]
R. W. Robinson, Tables
R. W. Robinson, Tables [Local copy, with permission]
R. W. Robinson, Enumeration of non-separable graphs, J. Combin. Theory 9 (1970), 327-356.
R. W. Robinson and T. R. S. Walsh, Inversion of cycle index sum relations for 2- and 3-connected graphs, J. Combin. Theory Ser. B. 57 (1993), 289-308.
R. W. Robinson and T. R. S. Walsh, Inversion of cycle index sum relations for 2- and 3-connected graphs, J. Combin. Theory Ser. B. 57 (1993), 289-308.
Andrés Santos, Density Expansion of the Equation of State, in A Concise Course on the Theory of Classical Liquids, Volume 923 of the series Lecture Notes in Physics, pp 33-96, 2016. DOI:10.1007/978-3-319-29668-5_3. See Reference 40.
Andrew J. Schultz and David A. Kofke, Fifth to eleventh virial coefficients of hard spheres, Phys. Rev. E 90, 023301, 4 August 2014
D. Stolee, Isomorph-free generation of 2-connected graphs with applications, arXiv preprint arXiv:1104.5261 [math.CO], 2011.
Rodrigo Stange Tessinari, Marcia Helena Moreira Paiva, Maxwell E. Monteiro, Marcelo E. V. Segatto, Anilton Garcia, George T. Kanellos, Reza Nejabati, and Dimitra Simeonidou, On the Impact of the Physical Topology on the Optical Network Performance, IEEE British and Irish Conference on Optics and Photonics (BICOP 2018), London.
Eric Weisstein's World of Mathematics, Biconnected Graph
Eric Weisstein's World of Mathematics, k-Connected Graph

Column k=0 of A325111 (for n>1).
Column sums of A339070.
Row sums of A339071.
The labeled version is A013922.
Cf. A000088 (graphs), A001349 (connected graphs), A006289, A006290, A004115 (rooted case), A010355 (by edges), A241767.

\\ See A004115 for graphsSeries and A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(g=graphsSeries(n), gc=sLog(g), gcr=sPoint(gc)); intformal(x*sSolve( sLog( gcr/(x*sv(1)) ), gcr ), sv(1)) + sSolve(subst(gc, sv(1), 0), gcr)}
{ my(N=12); Vec(OgfSeries(cycleIndexSeries(N)), -N) } \\ Andrew Howroyd, Dec 28 2020

More terms from Ronald C. Read. Robinson and Walsh list the first 26 terms.
a(1) changed from 0 to 1 by Eric W. Weisstein, Dec 07 2021
a(1) restored to 0 by Andrew Howroyd, Feb 26 2023

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.

A316652 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 2, 9, 69, 623, 7793, 110430, 1906317, 36833614, 816101825, 19925210834, 541363267613, 15997458049946, 515769374925576, 17905023985615254, 669030297769291562, 26689471638523499483, 1134895275721374771655, 51161002326406795249910, 2440166138715867838359915

Offset: 1

Gus Wiseman, Jul 09 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(3) = 9 trees:
(1(11)), (111),
(1(12)), (2(11)), (112),
(1(23)), (2(13)), (3(12)), (123).

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A181821, A292504, A304660.

Cf. A316651, A316653, A316654, A316655, A316656.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,4}]

\\ See A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n )); x*Ser(v)}
StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Jan 04 2021

Terms a(10) and beyond from Andrew Howroyd, Jan 04 2021

A318566 Number of non-isomorphic multiset partitions of multiset partitions of multisets of size n.

Original entry on oeis.org

1, 6, 21, 104, 452, 2335, 11992, 66810, 385101, 2336352, 14738380, 96831730, 659809115, 4657075074, 33974259046, 255781455848, 1984239830571, 15839628564349, 129951186405574, 1094486382191624, 9453318070371926, 83654146992936350, 757769011659766015, 7020652591448497490

Offset: 1

Gus Wiseman, Aug 29 2018

nonn

			Non-isomorphic representatives of the a(3) = 21 multiset partitions of multiset partitions:
  {{{1,1,1}}}
  {{{1,1,2}}}
  {{{1,2,3}}}
  {{{1},{1,1}}}
  {{{1},{1,2}}}
  {{{1},{2,3}}}
  {{{2},{1,1}}}
  {{{1},{1},{1}}}
  {{{1},{1},{2}}}
  {{{1},{2},{3}}}
  {{{1}},{{1,1}}}
  {{{1}},{{1,2}}}
  {{{1}},{{2,3}}}
  {{{2}},{{1,1}}}
  {{{1}},{{1},{1}}}
  {{{1}},{{1},{2}}}
  {{{1}},{{2},{3}}}
  {{{2}},{{1},{1}}}
  {{{1}},{{1}},{{1}}}
  {{{1}},{{1}},{{2}}}
  {{{1}},{{2}},{{3}}}

Cf. A001970, A007716, A050336, A050338, A255906, A269134, A317533, A317791, A318564, A318565.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
dubnorm[m_]:=First[Union[Table[Map[Sort,m/.Rule@@@Table[{Union[Flatten[m]][[i]],Union[Flatten[m]][[perm[[i]]]]},{i,Length[perm]}],{0,2}],{perm,Permutations[Union[Flatten[m]]]}]]];
Table[Length[Union[dubnorm/@Join@@mps/@Join@@mps/@strnorm[n]]],{n,5}]

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=sExp(symGroupSeries(n))); NumUnlabeledObjsSeq(sCartProd(A, sExp(A)-1))} \\ Andrew Howroyd, Dec 30 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2020

A330465 Number of non-isomorphic series-reduced rooted trees whose leaves are multisets with a total of n elements.

Original entry on oeis.org

1, 4, 14, 87, 608, 5573, 57876, 687938, 9058892, 130851823, 2048654450, 34488422057, 620046639452, 11839393796270, 238984150459124, 5079583100918338, 113299159314626360, 2644085918303683758, 64393240540265515110, 1632731130253043991252, 43013015553755764179000

Offset: 1

Gus Wiseman, Dec 21 2019

nonn

Also inequivalent leaf-colorings of phylogenetic rooted trees with n labels. A phylogenetic rooted tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.

			Non-isomorphic representatives of the a(3) = 14 trees:
  ((1)((1)(1)))  ((1)((1)(2)))  ((1)((2)(3)))  ((2)((1)(1)))
  ((1)(1)(1))    ((1)(1)(2))    ((1)(2)(3))    ((2)(1,1))
  ((1)(1,1))     ((1)(1,2))     ((1)(2,3))
  (1,1,1)        (1,1,2)        (1,2,3)

The version where leaves are atoms is A318231.
The case with sets as leaves is A330624.
The case with disjoint sets as leaves is A141268.
Labeled versions are A330467 (strongly normal) and A330469 (normal).
The singleton-reduced version is A330470.
Cf. A000311, A000669, A004114, A005804, A007716, A281118, A289501, A292504, A316651, A316652, A318812, A319312, A330471, A330474, A330625, A339645.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n), p=sEulerT(x*sv(1) + O(x*x^n))); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sEulerT(x*Ser(v[1..n])), n ) + polcoef(p,n)); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 13 2020

Terms a(7) and beyond from Andrew Howroyd, Dec 13 2020

A318565 Number of multiset partitions of multiset partitions of strongly normal multisets of size n.

Original entry on oeis.org

1, 6, 27, 169, 1029, 7817, 61006, 547537, 5202009, 54506262, 606311524, 7299051826, 92985064466, 1264720212352, 18137495642192, 275078184766323, 4379514178076452, 73235806332442156, 1280229713195027792, 23381809052104639236, 444740694108284116235, 8801030741502964613534

Offset: 1

Gus Wiseman, Aug 29 2018

nonn

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities.

			The a(2) = 6 multiset partitions of multiset partitions:
  {{{1,1}}}
  {{{1,2}}}
  {{{1},{1}}}
  {{{1},{2}}}
  {{{1}},{{1}}}
  {{{1}},{{2}}}

Cf. A001970, A007716, A050336, A255906, A269134, A317533, A317791, A318564, A318566.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Sum[Length[mps[m]],{m,Join@@mps/@strnorm[n]}],{n,6}]

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); StronglyNormalLabelingsSeq(sExp(sExp(A))-1)} \\ Andrew Howroyd, Dec 30 2020

Terms a(9) and beyond from Andrew Howroyd, Dec 30 2020

A318564 Number of multiset partitions of multiset partitions of normal multisets of size n.

Original entry on oeis.org

1, 6, 36, 274, 2408, 24440, 279172, 3542798, 49354816, 747851112, 12231881948, 214593346534, 4016624367288, 79843503990710, 1678916979373760, 37215518578700028, 866953456654946948, 21167221410812128266, 540346299720320080828, 14390314687100383124540, 399023209689817997883900

Offset: 1

Gus Wiseman, Aug 29 2018

nonn

A multiset is normal if it spans an initial interval of positive integers.

			The a(2) = 6 multiset partitions of multiset partitions:
  {{{1,1}}}
  {{{1,2}}}
  {{{1},{1}}}
  {{{1},{2}}}
  {{{1}},{{1}}}
  {{{1}},{{2}}}

Cf. A001970, A007716, A050336, A255906, A269134, A317533, A317791, A318566.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Sum[Length[mps[m]],{m,Join@@mps/@allnorm[n]}],{n,6}]

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NormalLabelingsSeq(sExp(sExp(A))-1)} \\ Andrew Howroyd, Jan 01 2021

Terms a(8) and beyond from Andrew Howroyd, Jan 01 2021

A320665 Number of non-isomorphic multiset partitions of weight n with no singletons or vertices that appear only once.

Original entry on oeis.org

1, 0, 1, 1, 5, 6, 27, 47, 169, 406, 1327, 3790, 12560, 39919, 136821, 470589, 1687981, 6162696, 23173374, 88981796, 349969596, 1405386733, 5764142220, 24111709328, 102825231702, 446665313598, 1975339030948, 8888051121242, 40667889052853, 189126710033882, 893526261542899

Offset: 0

Gus Wiseman, Oct 18 2018

nonn

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. This sequence counts non-isomorphic multiset partitions with no singletons whose dual also has no singletons.

			Non-isomorphic representatives of the a(2) = 1 through a(6) = 27 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}    {{1,1,1,1,1,1}}
                      {{1,1,2,2}}    {{1,1,2,2,2}}    {{1,1,1,2,2,2}}
                      {{1,1},{1,1}}  {{1,1},{1,1,1}}  {{1,1,2,2,2,2}}
                      {{1,1},{2,2}}  {{1,1},{1,2,2}}  {{1,1,2,2,3,3}}
                      {{1,2},{1,2}}  {{1,1},{2,2,2}}  {{1,1},{1,1,1,1}}
                                     {{1,2},{1,2,2}}  {{1,1,1},{1,1,1}}
                                                      {{1,1},{1,2,2,2}}
                                                      {{1,1,1},{2,2,2}}
                                                      {{1,1,2},{1,2,2}}
                                                      {{1,1},{2,2,2,2}}
                                                      {{1,1,2},{2,2,2}}
                                                      {{1,1},{2,2,3,3}}
                                                      {{1,1,2},{2,3,3}}
                                                      {{1,2},{1,1,2,2}}
                                                      {{1,2},{1,2,2,2}}
                                                      {{1,2},{1,2,3,3}}
                                                      {{1,2,2},{1,2,2}}
                                                      {{1,2,3},{1,2,3}}
                                                      {{2,2},{1,1,2,2}}
                                                      {{1,1},{1,1},{1,1}}
                                                      {{1,1},{1,2},{2,2}}
                                                      {{1,1},{2,2},{2,2}}
                                                      {{1,1},{2,2},{3,3}}
                                                      {{1,1},{2,3},{2,3}}
                                                      {{1,2},{1,2},{1,2}}
                                                      {{1,2},{1,2},{2,2}}
                                                      {{1,2},{1,3},{2,3}}

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

Row sums of A369287.

Cf. A001055, A007716, A306005, A317794, A318871, A320663, A320664, A321760.

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A-x*sv(1)), sExp(A-x*sv(1))))} \\ Andrew Howroyd, Jan 17 2023

Vec(G(20,1)) \\ G defined in A369287. - Andrew Howroyd, Jan 28 2024

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

A318285 Number of non-isomorphic multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 3, 7, 7, 7, 9, 11, 12, 16, 5, 15, 17, 22, 16, 29, 19, 30, 16, 21, 30, 23, 29, 42, 52, 56, 7, 47, 45, 57, 43, 77, 67, 77, 31, 101, 98, 135, 47, 85, 97, 176, 29, 66, 64, 118, 77, 231, 69, 97, 57, 181, 139, 297, 137, 385, 195, 166, 11, 162, 171, 490, 118

Offset: 1

Gus Wiseman, Aug 23 2018

nonn

			Non-isomorphic representatives of the a(12) = 9 multiset partitions of {1,1,2,3}:
  {{1,1,2,3}}
  {{1},{1,2,3}}
  {{2},{1,1,3}}
  {{1,1},{2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{2},{3},{1,1}}
  {{1},{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..130

Cf. A007716, A181821, A255906, A269134, A305936, A317533, A317791.

Cf. A318283, A318284, A318287, A318362, A318371.

\\ See links in A339645 for combinatorial species functions.
sig(n)={my(f=factor(n), sig=vector(primepi(vecmax(f[,1])))); for(i=1, #f~, sig[primepi(f[i,1])]=f[i,2]); sig}
C(sig)={my(n=sum(i=1, #sig, i*sig[i]), A=Vec(symGroupSeries(n)-1), B=O(x*x^n), c=prod(i=1, #sig, if(sig[i], sApplyCI(A[sig[i]], sig[i], A[i], i), 1))); polcoef(OgfSeries(sCartProd(c*x^n + B, sExp(x*Ser(A) + B))), n)}
a(n)={if(n==1, 1, C(sig(n)))} \\ Andrew Howroyd, Jan 17 2023

a(n) = A317791(A181821(n)).

Terms a(31) and beyond from Andrew Howroyd, Jan 17 2023

A318231 Number of inequivalent leaf-colorings of series-reduced rooted trees with n nodes.

Original entry on oeis.org

1, 0, 2, 3, 9, 23, 73, 229, 796, 2891, 11118, 44695, 187825, 820320, 3716501, 17413308, 84209071, 419461933, 2148673503, 11301526295, 60956491070, 336744177291, 1903317319015, 10995856040076, 64873456288903, 390544727861462, 2397255454976268, 14993279955728851

Offset: 1

Gus Wiseman, Aug 21 2018

nonn

In a series-reduced rooted tree, every non-leaf node has at least two branches.

			Inequivalent representatives of the a(6) = 23 leaf-colorings:
  (11(11))  (1(111))  (11111)
  (11(12))  (1(112))  (11112)
  (11(22))  (1(122))  (11122)
  (11(23))  (1(123))  (11123)
  (12(11))  (1(222))  (11223)
  (12(12))  (1(223))  (11234)
  (12(13))  (1(234))  (12345)
  (12(33))
  (12(34))

Cf. A000081, A001190, A001678, A003238, A004111, A290689, A291636, A304486.

Cf. A318226, A318227, A318228, A318229, A318230, A318234, A339645, A339648.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sEulerT(x*Ser(concat(v[1..n-2], [0]))), n-1 )); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 11 2020

cp's OEIS Frontend

A002218 Number of unlabeled nonseparable (or 2-connected) graphs (or blocks) with n nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A316652 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A318566 Number of non-isomorphic multiset partitions of multiset partitions of multisets of size n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A330465 Number of non-isomorphic series-reduced rooted trees whose leaves are multisets with a total of n elements.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A318565 Number of multiset partitions of multiset partitions of strongly normal multisets of size n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A318564 Number of multiset partitions of multiset partitions of normal multisets of size n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A320665 Number of non-isomorphic multiset partitions of weight n with no singletons or vertices that appear only once.

Views

Author