A000664 -id:A000664 - cp's OEIS frontend

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

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

A331508 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set multipartitions (multisets of sets) on a k-set.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 5, 3, 1, 1, 0, 1, 11, 12, 4, 1, 1, 0, 1, 26, 66, 25, 5, 1, 1, 0, 1, 68, 445, 278, 44, 6, 1, 1, 0, 1, 177, 4279, 5532, 966, 73, 7, 1, 1, 0, 1, 497, 53340, 200589, 53535, 2957, 112, 8, 1, 1, 0, 1, 1476, 846254, 11662671, 7043925, 431805, 8149, 166, 9, 1, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

An n-regular set multipartition is a finite multiset of nonempty sets in which each element appears in n blocks.
A set multipartition is T_0 if for every two distinct elements there exists a block containing one but not the other element.
A(n,k) is the number of nonequivalent binary matrices with k distinct columns and any number of nonzero rows with n ones in every column up to permutation of rows and columns.
A(n,k) is the number of non-isomorphic set-systems with k parts each of size n.

			Array begins:
===============================================
n\k | 0 1 2  3    4      5       6        7
----+------------------------------------------
  0 | 1 1 0  0    0      0       0        0 ...
  1 | 1 1 1  1    1      1       1        1 ...
  2 | 1 1 2  5   11     26      68      177 ...
  3 | 1 1 3 12   66    445    4279    53340 ...
  4 | 1 1 4 25  278   5532  200589 11662671 ...
  5 | 1 1 5 44  966  53535 7043925 ...
  6 | 1 1 6 73 2957 431805 ...
  ...
The A(2,3) = 5 matrices are:
  [1 0 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 0]
  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 1]  [1 0 1]
  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 1]
  [0 1 0]  [0 0 1]  [0 0 1]  [0 0 1]
  [0 0 1]  [0 0 1]
  [0 0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..152 (first 17 antidiagonals)

Rows n=1..6 are A000012, A000664, A058790, A058791, A058792, A058793.

Cf. A283877, A306019, A330942, A331126, A331277, A331461, A331509, A331510.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
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}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))[k]}
T(n,k)={my(m=n*k, s=0); if(m==0, k<=1, forpart(q=m, my(g=sum(t=1, k, K(q, t, n)*x^t/t) + O(x*x^k)); s+=permcount(q)*polcoef(exp(g - subst(g,x,x^2)), k)); s/m!)}
{ for(n=0, 6, for(k=0, 5, print1(T(n, k), ", ")); print) } \\ Andrew Howroyd, Jan 16 2024

A306019(n) = Sum_{d|n} A(n/d, d).

A053419 Number of graphs with loops (symmetric relations) with n edges.

Original entry on oeis.org

1, 2, 5, 14, 38, 107, 318, 972, 3111, 10410, 36371, 132656, 504636, 1998361, 8224448, 35112342, 155211522, 709123787, 3342875421, 16234342515, 81102926848, 416244824068, 2192018373522, 11831511359378, 65387590986455, 369661585869273, 2135966349269550, 12604385044890628

Offset: 0

Vladeta Jovovic, Jan 10 2000

nonn

In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second. a(n) is the number of non-isomorphic multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} with no equivalent vertices. For example, non-isomorphic representatives of the a(2) = 5 multiset partitions are (1)(122), (11)(22), (1)(1)(22), (1)(2)(12), (1)(1)(2)(2). - Gus Wiseman, Jul 18 2018

a(n) is the number of unlabeled simple graphs with n edges rooted at one vertex. - Andrew Howroyd, Nov 22 2020

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

Cf. A000664, A000666, A007716, A007717, A020555, A050535, A053419, A094574, A191970 (multisets), A316974, A339063.

seq[n_] := Module[{A = O[x]^n}, G[2n, x+A, {1}] // CoefficientList[#, x]&]; (* Jean-François Alcover, Dec 03 2020, using Andrew Howroyd's code for g.f. G in A339063 *)

\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1]))} \\ Andrew Howroyd, Nov 22 2020

Euler transform of A191970. - Andrew Howroyd, Oct 22 2019

a(16)-a(24) from Max Alekseyev, Jan 22 2010

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

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

A000608 Number of connected partially ordered sets with n unlabeled elements.

Original entry on oeis.org

1, 1, 1, 3, 10, 44, 238, 1650, 14512, 163341, 2360719, 43944974, 1055019099, 32664984238, 1303143553205, 66900392672168, 4413439778321689

Offset: 0

N. J. A. Sloane, Simon Plouffe, J. A. Wright

K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.
G. Melançon, personal communication.
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).

Gunnar Brinkmann and Brendan D. McKay, Counting unlabeled topologies and transitive relations.
G. Brinkmann and B. D. McKay, Counting unlabeled topologies and transitive relations, J. Integer Sequences, Volume 8, 2005.
G. Brinkmann and B. D. McKay, Posets on up to 16 Points [On Brendan McKay's home page]
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179.
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961. [Annotated scanned copy]
Henry Sharp, Jr., Quasi-orderings and topologies on finite sets, Proceedings of the American Mathematical Society 17.6 (1966): 1344-1349. [Annotated scanned copy]
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
N. J. A. Sloane, Transforms
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 10 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
N. L. White, Two letters to N. J. A. Sloane, 1970, with hand-drawn enclosure
Wikipedia, Illustration n=4, Illustration n=5, Illustration n=6" (2021)
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970. [Annotated scanned copy]
J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
J. A. Wright, Two related abstracts, 1970 and 1972 [Annotated scanned copies]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for sequences related to posets

Inverse Euler transform of A000112.

Cf. A263864 (multiset transform), A342500 (refined by rank).

A000112 = Cases[Import["https://oeis.org/A000112/b000112.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
{1} ~Join~ EulerInvTransform[Rest[A000112]] (* Jean-François Alcover, Dec 04 2019, updated Mar 17 2020 *)

More terms from Christian G. Bower, who pointed out connection with A000112, Jan 21 1998 and Dec 12 2001

More terms from Vladeta Jovovic, Jan 04 2006; corrected Jan 15 2006

A095133 Triangle of numbers of forests on n nodes containing k trees.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 6, 6, 4, 2, 1, 1, 11, 11, 7, 4, 2, 1, 1, 23, 23, 14, 8, 4, 2, 1, 1, 47, 46, 29, 15, 8, 4, 2, 1, 1, 106, 99, 60, 32, 16, 8, 4, 2, 1, 1, 235, 216, 128, 66, 33, 16, 8, 4, 2, 1, 1, 551, 488, 284, 143, 69, 34, 16, 8, 4, 2, 1, 1, 1301, 1121, 636, 315, 149, 70, 34, 16, 8, 4, 2, 1, 1

Offset: 1

Eric W. Weisstein, May 29 2004

Row sums are A005195.

For k > n/2, T(n,k) = T(n-1,k-1). - Geoffrey Critzer, Oct 13 2012

			Triangle begins:
    1;
    1,  1;
    1,  1,  1;
    2,  2,  1,  1;
    3,  3,  2,  1,  1;
    6,  6,  4,  2,  1, 1;
   11, 11,  7,  4,  2, 1, 1;
   23, 23, 14,  8,  4, 2, 1, 1;
   47, 46, 29, 15,  8, 4, 2, 1, 1;
  106, 99, 60, 32, 16, 8, 4, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 12 (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 6 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Forest

Cf. A005195 (row sums), A005196, A106240, A000055 (first column), A274937 (2nd column), A105821.
Limiting sequence of reversed rows gives A215930.
Reflected table is A136605. - Alois P. Heinz, Apr 11 2014

with(numtheory):
b:= proc(n) option remember; local d, j; `if` (n<=1, n,
      (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
    end:
t:= proc(n) option remember; local k; `if` (n=0, 1,
      b(n)-(add(b(k)*b(n-k), k=0..n)-`if`(irem(n, 2)=0, b(n/2), 0))/2)
    end:
g:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(g(n-i*j, i-1, p-j) *
       binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
    end:
a:= (n, k)-> g(n, n, k):
seq(seq(a(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 20 2012

nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i]s[n-1,i]i,{i,1,n-1}]/(n-1);ft=Table[a[i]-Sum[a[j]a[i-j],{j,1,i/2}]+If[OddQ[i],0,a[i/2](a[i/2]+1)/2],{i,1,nn}];CoefficientList[Series[Product[1/(1-y x^i)^ft[[i]],{i,1,nn}],{x,0,20}],{x,y}]//Grid (* Geoffrey Critzer, Oct 13 2012, after code given by Robert A. Russell in A000055 *)

T(n, k) = sum over the partitions of n, 1M1 + 2M2 + ... + nMn, with exactly k parts, of Product_{i=1..n} binomial(A000055(i) + Mi - 1, Mi). - Washington Bomfim, May 12 2005

More terms from Vladeta Jovovic, Jun 03 2004

A027416 Number of unlabeled (and unrooted) trees on n nodes having a centroid.

Original entry on oeis.org

1, 1, 0, 1, 1, 3, 3, 11, 13, 47, 61, 235, 341, 1301, 1983, 7741, 12650, 48629, 82826, 317955, 564225, 2144505, 3926353, 14828074, 27940136, 104636890, 201837109, 751065460, 1479817181, 5469566585, 10975442036, 40330829030, 82270184950

Offset: 0

N. J. A. Sloane

nonn

Also, number of rooted unlabeled trees on n nodes not having a primary branch.
A tree has either a center or a bicenter and either a centroid or a bicentroid. (These terms were introduced by Jordan.)
If the number of edges in a longest path in the tree is 2m, then the middle node in the path is the unique center, otherwise the two middle nodes in the path are the unique bicenters.
On the other hand, define the weight of a node P to be the greatest number of nodes in any subtree connected to P. Then either there is a unique node of minimal weight, the centroid of the tree, or there is a unique pair of minimal weight nodes, the bicentroids.
Let T be a tree with root node R. If R and the edges incident with it are deleted, the resulting rooted trees are called branches. A primary branch (there can be at most one) has i nodes where n/2 <= i <= n-1.

F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1994; pp. 35, 36.

N. J. A. Sloane, Table of n, a(n) for n = 0..200
A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268.
C. Jordan, Sur les assemblages des lignes, J. Reine angew. Math., 70 (1869), 185-190.
A. Meir and J. W. Moon, On the branch-sizes of rooted unlabeled trees, in "Graph Theory and Its Applications", Annals New York Acad. Sci., Vol. 576, 1989, pp. 399-407.
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1. [This articles states incorrectly that A000676 and A000677 give the numbers of trees with respectively a centroid and bicentroid.]
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 12 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Index entries for sequences related to rooted trees

Cf. A102911 (trees with a bicentroid), A027415 (trees with a primary branch), A000676 (trees with a center), A000677 (trees with a bicenter), A000055 (trees), A000081 (rooted trees).

N := 50: Y := [ 1,1 ]: for n from 3 to N do x*mul( (1-x^i)^(-Y[ i ]), i=1..n-1); series(%,x,n+1); b := coeff(%,x,n); Y := [ op(Y),b ]; od: P:=n->sum(Y[n-i]*Y[i],i=1..floor(n/2)): seq(Y[n]-P(n),n=1..35); # Emeric Deutsch, Nov 21 2004

a(n) = A000055(n) - A102911(n/2) if n is even, else a(n) = A000055(n).
a(n) = A000081(n) - A027415(n). - Emeric Deutsch, Nov 21 2004
a(n) = [x^n] 1 + x/Product_{i=1..ceiling(n/2)-1} (1-x^i)^A000081(i).  See Cayley link above. - Geoffrey Critzer, Jul 30 2022

More terms from Emeric Deutsch, Nov 21 2004

Entry revised (with new definition) by N. J. A. Sloane, Feb 26 2007

A339063 Number of unlabeled simple graphs with n edges rooted at two noninterchangeable vertices.

Original entry on oeis.org

1, 4, 13, 43, 141, 467, 1588, 5544, 19966, 74344, 286395, 1141611, 4707358, 20063872, 88312177, 400980431, 1875954361, 9032585846, 44709095467, 227245218669, 1184822316447, 6330552351751, 34630331194626, 193785391735685, 1108363501628097, 6474568765976164

Offset: 0

Andrew Howroyd, Nov 22 2020

nonn

			The a(1) = 4 cases correspond to a single edge which can be attached to zero, one or both of the roots.

Cf. A000664, A053419 (one root), A304070, A339040, A339064, 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];
edges[v_, t_] := Product[With[{g = GCD[v[[i]], v[[j]]]}, t[v[[i]]*v[[j]]/ g]^g], {i, 2, Length[v]}, {j, 1, i-1}]*Product[With[{c = v[[i]]}, t[c]^Quotient[c-1, 2]*If[OddQ[c], 1, t[c/2]]], {i, 2, Length[v]}];
G[n_, x_, r_] := Module[{s = 0}, Do[s += permcount[p]*edges[Join[r, p], 1+x^#&], {p, IntegerPartitions[n]}]; s/n!];
seq[n_] := Module[{A = O[x]^n}, G[2n, x+A, {1, 1}]//CoefficientList[#, x]&];
seq[15] (* Jean-François Alcover, Dec 03 2020, 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, x, r)={my(s=0); forpart(p=n, s+=permcount(p)*edges(concat(r, Vec(p)), i->1+x^i)); s/n!}
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1])))}

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cp's OEIS Frontend

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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A000676 Number of centered trees with n nodes.

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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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A331508 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set multipartitions (multisets of sets) on a k-set.

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A053419 Number of graphs with loops (symmetric relations) with n edges.

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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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A000608 Number of connected partially ordered sets with n unlabeled elements.

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A095133 Triangle of numbers of forests on n nodes containing k trees.

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