A007719 -id:A007719 - cp's OEIS frontend

A007717 Number of symmetric polynomial functions of degree n of a symmetric matrix (of indefinitely large size) under joint row and column permutations. Also number of multigraphs with n edges (allowing loops) on an infinite set of nodes.

1, 2, 7, 23, 79, 274, 1003, 3763, 14723, 59663, 250738, 1090608, 4905430, 22777420, 109040012, 537401702, 2723210617, 14170838544, 75639280146, 413692111521, 2316122210804, 13261980807830, 77598959094772, 463626704130058, 2826406013488180, 17569700716557737

Offset: 0

Colin Mallows

nonn

Euler transform of A007719.
Also the number of non-isomorphic multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018
Number of distinct n X 2n matrices with integer entries and rows sums 2, up to row and column permutations. - Andrew Howroyd, Sep 06 2018
a(n) is the number of unlabeled loopless multigraphs with n edges rooted at one vertex. - Andrew Howroyd, Nov 22 2020

			a(2) = 7 (here - denotes an edge, = denotes a pair of parallel edges and o is a loop):
  oo
  o o
  o-
  o -
  =
  --
  - -
From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(2) = 7 multiset partitions of {1, 1, 2, 2}:
  (1122),
  (1)(122), (11)(22), (12)(12),
  (1)(1)(22), (1)(2)(12),
  (1)(1)(2)(2).
(End)
From _Gus Wiseman_, Jan 08 2024: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(3) = 7 rooted loopless multigraphs (root shown as singleton):
  {{1}}  {{1},{1,2}}  {{1},{1,2},{1,2}}
         {{1},{2,3}}  {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{1,2},{3,4}}
                      {{1},{2,3},{2,3}}
                      {{1},{2,3},{2,4}}
                      {{1},{2,3},{4,5}}
(End)

Huaien Li and David C. Torney, Enumerations of Multigraphs, 2002.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Mateo Díaz, Dmitriy Drusvyatskiy, Jack Kendrick, and Rekha R. Thomas, Invariant Kernels: Rank Stabilization and Generalization Across Dimensions, arXiv:2502.01886 [math.OC], 2025. See p. 43.
Huaien Li and David C. Torney, Enumeration of unlabelled multigraphs, Ars Combin. 75 (2005) 171-188. MR2133219.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) table 67.

Row n=2 of A331485.

Cf. A000664, A002620, A007716, A007719, A020555, A050531, A050532, A050535, A052171, A053418, A053419, A094574, A316972, A316974, A318951, A339065.

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!];
a[n_] := RowSumMats[n, 2n, 2];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 25}] (* Jean-François Alcover, Oct 27 2018, after Andrew Howroyd *)

\\ See A318951 for RowSumMats
a(n)=RowSumMats(n, 2*n, 2); \\ Andrew Howroyd, Sep 06 2018

\\ See A339065 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1]))} \\ Andrew Howroyd, Nov 22 2020

More terms from Vladeta Jovovic, Jan 26 2000

a(0)=1 prepended and a(16)-a(25) added by Max Alekseyev, Jun 21 2011

A322114 Regular triangle read by rows where T(n,k) is the number of unlabeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 3, 2, 0, 0, 3, 6, 3, 0, 0, 2, 11, 14, 6, 0, 0, 1, 13, 35, 33, 11, 0, 0, 0, 10, 61, 112, 81, 23, 0, 0, 0, 5, 75, 262, 347, 204, 47, 0, 0, 0, 2, 68, 463, 1059, 1085, 526, 106, 0, 0, 0, 1, 49, 625, 2458, 4091, 3348, 1376, 235

Offset: 0

Gus Wiseman, Nov 26 2018

			Triangle begins:
   1
   1   1
   0   1   1
   0   1   3   2
   0   0   3   6   3
   0   0   2  11  14   6
   0   0   1  13  35  33  11
Non-isomorphic representatives of the graphs counted in row 4:
  {{2}{3}{12}{13}}   {{4}{12}{23}{34}}   {{13}{24}{35}{45}}
  {{2}{3}{13}{23}}   {{4}{13}{23}{34}}   {{14}{25}{35}{45}}
  {{3}{12}{13}{23}}  {{4}{13}{24}{34}}   {{15}{25}{35}{45}}
                     {{4}{14}{24}{34}}
                     {{12}{13}{24}{34}}
                     {{14}{23}{24}{34}}

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

Row sums are A191970. Last column is A000055.

Cf. A000664, A007716, A007718, A007719, A054923, A191646, A275421, A317533, A321254.

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 22 2019

Terms a(28) and beyond from Andrew Howroyd, Oct 22 2019

A316974 Number of non-isomorphic strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.

Original entry on oeis.org

1, 1, 4, 14, 49, 173, 652, 2494

Offset: 0

Gus Wiseman, Jul 17 2018

Also the number of unlabeled multigraphs with n edges, allowing loops, spanning an initial interval of positive integers with no equivalent vertices (two vertices are equivalent if in every edge the multiplicity of the first is equal to the multiplicity of the second). For example, non-isomorphic representatives of the a(2) = 4 multigraphs are {(1,2),(1,3)}, {(1,1),(1,2)}, {(1,1),(2,2)}, {(1,1),(1,1)}.

			Non-isomorphic representatives of the a(3) = 14 strict multiset partitions:
  (112233),
  (1)(12233), (11)(2233), (12)(1233), (112)(233),
  (1)(2)(1233), (1)(12)(233), (1)(23)(123), (2)(11)(233), (11)(22)(33), (12)(13)(23),
  (1)(2)(3)(123), (1)(2)(12)(33), (1)(2)(13)(23).

Cf. A001055, A007716, A007717, A007719, A020554, A020555, A045778, A050535, A053419, A061742, A094574, A162247, A316892, A316972.

a(7) from Andrew Howroyd, Feb 07 2020

A322115 Triangle read by rows where T(n,k) is the number of unlabeled connected multigraphs with loops with n edges and k vertices.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 2, 1, 6, 11, 9, 3, 1, 9, 25, 34, 20, 6, 1, 12, 52, 104, 99, 49, 11, 1, 16, 94, 274, 387, 298, 118, 23, 1, 20, 162, 645, 1295, 1428, 881, 300, 47, 1, 25, 263, 1399, 3809, 5803, 5088, 2643, 765, 106, 1, 30, 407, 2823, 10187, 20645, 24606, 17872, 7878, 1998, 235

Offset: 0

Gus Wiseman, Nov 26 2018

			Triangle begins:
  1
  1   1
  1   2   1
  1   4   4   2
  1   6  11   9   3
  1   9  25  34  20   6
  1  12  52 104  99  49  11

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Row sums are A007719. Diagonal k = n-1 is A000055.

Cf. A000664, A007716, A007718, A191646, A191970, A275421, A317533, A322114.

EulerT(v)={my(p=exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1); Vec(p/x,-#v)}
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,x)={sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i],v[j])); g*x^(v[i]*v[j]/g))) + sum(i=1, #v, my(t=v[i]); ((t+1)\2)*x^t + if(t%2, 0, x^(t/2)))}
G(n,m)={my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(edges(p,x) + O(x*x^m), -m))); s/n!}
R(n)={Mat(apply(p->Col(p+O(y^n), -n), InvEulerMT(vector(n, k, 1 + y*Ser(G(k,n-1), y)))))}
{ my(T=R(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Nov 30 2018

Terms a(28) and beyond from Andrew Howroyd, Nov 30 2018

A036250 Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 35, 85, 231, 633, 1845, 5461, 16707, 51945, 164695, 529077, 1722279, 5664794, 18813369, 62996850, 212533226, 721792761, 2466135375, 8471967938, 29249059293, 101440962296, 353289339927, 1235154230060, 4333718587353, 15255879756033

Offset: 0

Christian G. Bower, Nov 15 1998

nonn

Also the number of non-isomorphic connected multigraphs with loops with n edges and multiset density -1, where the multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices. - Gus Wiseman, Nov 28 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1717
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Gus Wiseman, Non-isomorphic representatives of the a(1) = 2 through a(5) = 35 connected multigraphs with loops with multiset density -1.
Index entries for sequences related to trees

Essentially the same as A036251.

Cf. A000055, A007718, A007719, A038052, A191646, A303837, A321155, A321229, A321254, A321256, A322111.

max = 30; B[] = 1; Do[B[x] = x*Exp[Sum[(B[x^k] + x^k)/k + O[x]^n, {k, 1, n}]] // Normal, {n, 1, max}]; A[x_] = B[x] - B[x]^2/2 + B[x^2]/2; CoefficientList[1 + A[x] + O[x]^max, x] (* Jean-François Alcover, Jan 28 2019 *)

G.f.: B(x) - B^2(x)/2 + B(x^2)/2, where B(x) is g.f. for A036249.

A316972 Number of connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.

Original entry on oeis.org

1, 2, 5, 28, 277, 3985, 76117, 1833187, 53756682, 1871041538, 75809298105, 3521419837339, 185235838688677, 10923147890901151, 715989783027216302, 51793686238309903860, 4109310551278549543317, 355667047514571431358297, 33422937748872646130124797

Offset: 0

Gus Wiseman, Jul 17 2018

nonn

Note that all connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} are strict except for (123...n)(123...n).

			The a(2) = 5 connected multiset partitions of {1, 1, 2, 2} are (1122), (1)(122), (2)(112), (12)(12), (1)(2)(12). The multiset partitions (11)(22), (1)(1)(22), (2)(2)(11), (1)(1)(2)(2) are not connected.

Cf. A001055, A007716, A007717, A007719, A020555, A045778, A061742, A094574, A162247, A316974.

nn=10;
ser=Exp[-3/2+Exp[x]/2]*Sum[Exp[Binomial[n+1,2]*x]/n!,{n,0,3*nn}];
Round/@(CoefficientList[Series[1+Log[ser],{x,0,nn}],x]*Array[Factorial,nn+1,0]) (* based on Jean-François Alcover after Vladeta Jovovic *)
(*second program *)
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]]]]]]]]];
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]]]];
Length/@Table[Select[mps[Ceiling[Range[1/2,n,1/2]]],Length[csm[#]]==1&],{n,4}]

Logarithmic transform of A020555.

A322147 Regular triangle read by rows where T(n,k) is the number of labeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.

Original entry on oeis.org

1, 1, 1, 0, 2, 3, 0, 1, 10, 16, 0, 0, 12, 79, 125, 0, 0, 6, 162, 847, 1296, 0, 0, 1, 179, 2565, 11436, 16807, 0, 0, 0, 116, 4615, 47100, 185944, 262144, 0, 0, 0, 45, 5540, 121185, 987567, 3533720, 4782969, 0, 0, 0, 10, 4720, 220075, 3376450, 23315936, 76826061, 100000000

Offset: 0

Gus Wiseman, Nov 28 2018

			Triangle begins:
  1
  1     1
  0     2     3
  0     1    10    16
  0     0    12    79   125
  0     0     6   162   847  1296
  0     0     1   179  2565 11436 16807

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Row sums are A322151. Last column is A000272.
Column sums are A062740.
Cf. A000664, A007718, A007719, A054923, A191646, A275421, A321254, A322114, A322115, A322137.

multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
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[If[n==0,1,Length[Select[Subsets[multsubs[Range[k],2],{n}],And[Union@@#==Range[k],Length[csm[#]]==1]&]]],{n,0,6},{k,1,n+1}]

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}
M(n)={Mat([Col(p, -(n+1)) | p<-Connected(vector(2*n, j, (1 + x + O(x*x^n) )^binomial(j+1,2)))[1..n+1]])}
{ my(T=M(10)); for(n=1, #T, print(T[n,][1..n])) } \\ Andrew Howroyd, Nov 29 2018

Terms a(28) and beyond from Andrew Howroyd, Nov 29 2018

A322152 Number of labeled connected multigraphs with loops with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 2, 7, 39, 314, 3359, 45000, 725269, 13670256, 295099184, 7179749707, 194399095705, 5797793490859, 188855813757729, 6671188010874785, 254007814638737649, 10370334196814589256, 451923738493729293016, 20937747226064522726151, 1027666505638118490940059

Offset: 0

Gus Wiseman, Nov 28 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row sums of A322148. The unlabeled version is A007719.

Cf. A000272, A000664, A007718, A191646, A191970, A322114, A322115, A322147, A322151.

multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
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[multsubs[multsubs[Range[n+1],2],n],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,5}]

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/(1 - x + O(x*x^n))^binomial(j+1,2)))))} \\ Andrew Howroyd, Nov 28 2018

Terms a(7) and beyond from Andrew Howroyd, Nov 28 2018

A289988 Number of unlabeled connected loopless multigraphs with n nodes of degree n or less.

Original entry on oeis.org

1, 1, 2, 4, 37, 602, 34126, 6021463, 3616906549, 7361925161868, 51324462383008758, 1240420936122453106498, 105141919479926837860474091, 31581183353539008502807807352728

Offset: 0

Natan Arie Consigli, Aug 19 2017

Multigraphs are loopless.

Main diagonal of A334546.

Cf. A007719, A076864, A134818, A289157, A289158, A289987.

for n in {1..8}; do geng -c -D${n} ${n} -q | multig -m$[${n}-1] -D$[${n}-1] -u; done

a(0) corrected and a(9)-a(13) from Andrew Howroyd, May 05 2020

A321270 Number of connected multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 5, 4, 7, 3, 11, 7, 10, 1, 15, 9, 22, 7, 19, 12, 30, 5, 22, 19, 28, 14, 42, 22, 56, 1, 33, 30, 42, 20, 77, 45

Offset: 1

Gus Wiseman, Nov 01 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(2) = 1 through a(12) = 3 connected multiset partitions:
  {{1}}  {{11}}    {{12}}  {{111}}      {{112}}    {{1111}}
         {{1}{1}}          {{1}{11}}    {{1}{12}}  {{1}{111}}
                           {{1}{1}{1}}             {{11}{11}}
                                                   {{1}{1}{11}}
                                                   {{1}{1}{1}{1}}
.
  {{123}}  {{1122}}      {{1112}}      {{11111}}          {{1123}}
           {{1}{122}}    {{1}{112}}    {{1}{1111}}        {{1}{123}}
           {{12}{12}}    {{11}{12}}    {{11}{111}}        {{12}{13}}
           {{2}{112}}    {{1}{1}{12}}  {{1}{1}{111}}
           {{1}{2}{12}}                {{1}{11}{11}}
                                       {{1}{1}{1}{11}}
                                       {{1}{1}{1}{1}{1}}
The a(18) = 9, a(27) = 28, and a(36) = 20 connected multiset partitions of {1,1,2,2,3}, {1,1,2,2,3,3}, and {1,1,2,2,3,4} respectively:
  {{1,1,2,2,3}}      {{1,1,2,2,3,3}}        {{1,1,2,2,3,4}}
  {{1},{1,2,2,3}}    {{1},{1,2,2,3,3}}      {{1},{1,2,2,3,4}}
  {{1,2},{1,2,3}}    {{1,1,2},{2,3,3}}      {{1,1,2},{2,3,4}}
  {{1,3},{1,2,2}}    {{1,1,3},{2,2,3}}      {{1,2},{1,2,3,4}}
  {{2},{1,1,2,3}}    {{1,2},{1,2,3,3}}      {{1,2,2},{1,3,4}}
  {{2,3},{1,1,2}}    {{1,2,2},{1,3,3}}      {{1,2,3},{1,2,4}}
  {{1},{1,2},{2,3}}  {{1,2,3},{1,2,3}}      {{1,3},{1,2,2,4}}
  {{1},{2},{1,2,3}}  {{1,3},{1,2,2,3}}      {{1,4},{1,2,2,3}}
  {{2},{1,2},{1,3}}  {{2},{1,1,2,3,3}}      {{2},{1,1,2,3,4}}
                     {{2,3},{1,1,2,3}}      {{2,3},{1,1,2,4}}
                     {{3},{1,1,2,2,3}}      {{2,4},{1,1,2,3}}
                     {{1},{1,2},{2,3,3}}    {{1},{1,2},{2,3,4}}
                     {{1},{1,3},{2,2,3}}    {{1},{2},{1,2,3,4}}
                     {{1},{2},{1,2,3,3}}    {{1,2},{1,3},{2,4}}
                     {{1,2},{1,3},{2,3}}    {{1,2},{1,4},{2,3}}
                     {{1},{2,3},{1,2,3}}    {{1},{2,3},{1,2,4}}
                     {{1},{3},{1,2,2,3}}    {{1},{2,4},{1,2,3}}
                     {{2},{1,2},{1,3,3}}    {{2},{1,2},{1,3,4}}
                     {{2},{1,3},{1,2,3}}    {{2},{1,3},{1,2,4}}
                     {{2},{2,3},{1,1,3}}    {{2},{1,4},{1,2,3}}
                     {{2},{3},{1,1,2,3}}
                     {{3},{1,2},{1,2,3}}
                     {{3},{1,3},{1,2,2}}
                     {{3},{2,3},{1,1,2}}
                     {{1},{2},{1,3},{2,3}}
                     {{1},{2},{3},{1,2,3}}
                     {{1},{3},{1,2},{2,3}}
                     {{2},{3},{1,2},{1,3}}

Cf. A007718, A007719, A056156, A181821, A191970, A300913, A305193, A305936, A318284, A318286, A319557, A321272.

cp's OEIS Frontend

A007717 Number of symmetric polynomial functions of degree n of a symmetric matrix (of indefinitely large size) under joint row and column permutations. Also number of multigraphs with n edges (allowing loops) on an infinite set of nodes.

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A322114 Regular triangle read by rows where T(n,k) is the number of unlabeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.

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A316974 Number of non-isomorphic strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.

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A322115 Triangle read by rows where T(n,k) is the number of unlabeled connected multigraphs with loops with n edges and k vertices.

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PARI

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A036250 Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

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A316972 Number of connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.

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Mathematica

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A322147 Regular triangle read by rows where T(n,k) is the number of labeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.

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Mathematica

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A322152 Number of labeled connected multigraphs with loops with n edges (the vertices are {1,2,...,k} for some k).

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