A317672 -id:A317672 - cp's OEIS frontend

A323818 Number of connected set-systems covering n vertices.

1, 1, 4, 96, 31840, 2147156736, 9223372011084915712, 170141183460469231602560095199828453376, 57896044618658097711785492504343953923912733397452774312021795134847892828160

Offset: 0

Gus Wiseman, Jan 30 2019

nonn

Unlike the nearly identical sequence A092918, this sequence does not count under a(1) the a single-vertex hypergraph with no edges.

			The a(2) = 4 set-systems:
  {{1, 2}}
  {{1}, {1,2}}
  {{2}, {1,2}}
  {{1}, {2}, {1,2}}

G. C. Greubel, Table of n, a(n) for n = 0..11

Cf. A001187, A003465 (not necessarily connected), A016031, A048143, A092918, A293510, A317672, A323816, A323817 (no singletons), A323819 (unlabeled case).

m:=12;
f:= func< x | 1-x + Log( (&+[2^(2^n-1)*x^n/Factorial(n): n in [0..m+2]]) ) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Oct 04 2022

b:= n-> add(binomial(n, k)*2^(2^(n-k)-1)*(-1)^k, k=0..n):
a:= proc(n) option remember; b(n)-`if`(n=0, 0, add(
       k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n)
    end:
seq(a(n), n=0..8);  # Alois P. Heinz, Jan 30 2019

nn=8;
ser=Sum[2^(2^n-1)*x^n/n!,{n,0,nn}];
Table[SeriesCoefficient[1-x+Log[ser],{x,0,n}]*n!,{n,0,nn}]

m=12;
def f(x): return 1-x + log(sum(2^(2^n-1)*x^n/factorial(n) for n in range(m+2)))
def A_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A_list(m) # G. C. Greubel, Oct 04 2022

E.g.f.: 1 - x + log(Sum_{n >= 0} 2^(2^n-1) * x^n/n!).

Logarithmic transform of A003465.

A191646 Triangle read by rows: T(n,k) = number of connected multigraphs with n >= 0 edges and 1 <= k <= n+1 vertices, with no loops allowed.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 11, 11, 6, 0, 1, 6, 22, 34, 29, 11, 0, 1, 7, 37, 85, 110, 70, 23, 0, 1, 9, 61, 193, 348, 339, 185, 47, 0, 1, 11, 95, 396, 969, 1318, 1067, 479, 106, 0, 1, 13, 141, 771, 2445, 4457, 4940, 3294, 1279, 235

Offset: 0

Alberto Tacchella, Jul 04 2011

			Triangle T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 2,  2;
  0, 1, 3,  5,  3;
  0, 1, 4, 11, 11,  6;
  0, 1, 6, 22, 34, 29, 11;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274 (terms 0..119 from R. J. Mathar)
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017; see Section 4.
Brendan McKay and Adolfo Piperno, nauty and Traces. [nauty and Traces are programs for computing automorphism groups of graphs and digraphs.]
B. D. McKay and A. Piperno, Practical Graph Isomorphism, II, J. Symbolic Computation 60 (2013), 94-112.
Gordon Royle, Small Multigraphs.
Gus Wiseman, Illustration of the 33 connected multigraphs counted in row 5.

Row sums give A076864. Diagonal is A000055.
Cf. A034253, A054923, A192517, A253186 (column k=3), A290778 (column k=4).
Cf. A000664, A007718, A036250, A050535, A191646, A191970, A275421, A317672, A322114, A322133, A322152.

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(A=R(10)); for(n=1, #A, for(k=1, n, print1(A[n,k], ", "));print) } \\ Andrew Howroyd, May 14 2018

T(n,k=3) = A253186(n) = A034253(n,k=2) for n >= 1. - Petros Hadjicostas, Oct 02 2019

A054923 Triangle read by rows: number of connected graphs with k >= 0 edges and n nodes (1<=n<=k+1).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 3, 0, 0, 0, 1, 5, 6, 0, 0, 0, 1, 5, 13, 11, 0, 0, 0, 0, 4, 19, 33, 23, 0, 0, 0, 0, 2, 22, 67, 89, 47, 0, 0, 0, 0, 1, 20, 107, 236, 240, 106, 0, 0, 0, 0, 1, 14, 132, 486, 797, 657, 235, 0, 0, 0, 0, 0, 9, 138, 814, 2075, 2678, 1806, 551, 0, 0, 0, 0, 0, 5, 126, 1169, 4495, 8548, 8833, 5026, 1301

Offset: 0

N. J. A. Sloane

The diagonal n = k+1 is A000055(n). - Jonathan Vos Post, Aug 10 2008

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 2;
  0, 0, 0, 2, 3;
  0, 0, 0, 1, 5   6;
  0, 0, 0, 1, 5, 13,  11;
  0, 0, 0, 0, 4, 19,  33,  23;
  0, 0, 0, 0, 2, 22,  67,  89,  47;
  0, 0, 0, 0, 1, 20, 107, 236, 240, 106;
  ... (so with 5 edges there's 1 graph with 4 nodes, 5 with 5 nodes and 6 with 6 nodes). [Typo corrected by Anders Haglund, Jul 08 2008]

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 93, Table 4.2.2; p. 241, Table A2.

Andrew Howroyd, Table of n, a(n) for n = 0..1325
G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017), Table 57.
Gordon Royle, Small graphs
Gus Wiseman, Non-isomorphic representatives of the 12 connected graphs counted in row 5.

Main diagonal is A000055.
Subsequent diagonals give the number of connected unlabeled graphs with n nodes and n+k edges for k=0..2: A001429, A001435, A001436.
Cf. A002905 (row sums), A001349 (column sums), A008406, A046751 (transpose), A054924 (transpose), A046742 (w/o left column), A343088 (labeled).
Cf. A000664, A007718, A050535, A054923, A191646, A191970, A317672, A322114, A322151.

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 23 2019

a(83)-a(89) corrected by Andrew Howroyd, Oct 24 2019

A076864 Number of connected loopless multigraphs with n edges.

Original entry on oeis.org

1, 1, 2, 5, 12, 33, 103, 333, 1183, 4442, 17576, 72810, 314595, 1410139, 6541959, 31322474, 154468852, 783240943, 4077445511, 21765312779, 118999764062, 665739100725, 3807640240209, 22246105114743, 132672322938379, 807126762251748

Offset: 0

N. J. A. Sloane, Nov 23 2002

Inverse Euler transform of A050535.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Patrick T. Komiske, Eric M. Metodiev, and Jesse Thaler, Energy flow polynomials: A complete linear basis for jet substructure, arXiv:1712.07124 [hep-ph], 2017.
Tsuyoshi Miezaki, Akihiro Munemasa, Yusaku Nishimura, Tadashi Sakuma, and Shuhei Tsujie, Universal graph series, chromatic functions, and their index theory, arXiv:2403.09985 [math.CO], 2024. See p. 23.
N. J. A. Sloane, Transforms
Gus Wiseman, Non-isomorphic representatives of the unlabeled connected multigraphs counted by the first 5 terms

Row sums of A191646.
Cf. A050535 (multisets), A076865, A076866, A076867.
Cf. A000664, A007718, A054923, A191970, A275421, A317672, A322114, A322137, A322148.

A050535 = Cases[Import["https://oeis.org/A050535/b050535.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
Join[{1}, EulerInvTransform[A050535 // Rest]] (* Jean-François Alcover, Feb 11 2020, updated Mar 17 2020 *)

More terms from Sean A. Irvine, Oct 02 2011
Name and comment swapped by Gus Wiseman, Nov 28 2018
a(0)=1 prepended by Andrew Howroyd, Oct 23 2019

A321194 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of weight n with k connected components.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 17, 12, 3, 1, 40, 35, 12, 3, 1, 125, 112, 45, 12, 3, 1, 354, 347, 148, 45, 12, 3, 1, 1159, 1122, 512, 163, 45, 12, 3, 1, 3774, 3651, 1724, 572, 163, 45, 12, 3, 1, 13113, 12320, 5937, 2020, 593, 163, 45, 12, 3, 1, 46426, 42407, 20492, 7117, 2110, 593, 163, 45, 12, 3, 1

Offset: 1

Gus Wiseman, Oct 29 2018

			Triangle begins:
      1
      3     1
      6     3     1
     17    12     3     1
     40    35    12     3     1
    125   112    45    12     3     1
    354   347   148    45    12     3     1
   1159  1122   512   163    45    12     3     1
   3774  3651  1724   572   163    45    12     3     1
  13113 12320  5937  2020   593   163    45    12     3     1
The fourth row counts the following non-isomorphic multiset partitions.
  {{1,1,1,1}}        {{1,1},{2,2}}      {{1},{2},{3,3}}    {{1},{2},{3},{4}}
  {{1,1,2,2}}        {{1},{2,2,2}}      {{1},{2},{3,4}}
  {{1,2,2,2}}        {{1},{2,3,3}}      {{1},{2},{3},{3}}
  {{1,2,3,3}}        {{1,2},{3,3}}
  {{1,2,3,4}}        {{1},{2,3,4}}
  {{1},{1,1,1}}      {{1,2},{3,4}}
  {{1,1},{1,1}}      {{1},{1},{2,2}}
  {{1},{1,2,2}}      {{1},{1},{2,3}}
  {{1,2},{1,2}}      {{1},{2},{2,2}}
  {{1,2},{2,2}}      {{1},{3},{2,3}}
  {{1,3},{2,3}}      {{1},{1},{2},{2}}
  {{2},{1,2,2}}      {{1},{2},{2},{2}}
  {{3},{1,2,3}}
  {{1},{1},{1,1}}
  {{1},{2},{1,2}}
  {{2},{2},{1,2}}
  {{1},{1},{1},{1}}

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

First column is A007718. Row sums are A007716.

Cf. A006126, A048143, A143543, A306005, A316983, A317672, A317674, A319616, A321155.

O.g.f.: Product 1/(1 - t*x^n)^A007718(n).

Terms a(56) and beyond from Andrew Howroyd, Jan 11 2024

A317674 Regular triangle where T(n,k) is the number of antichains covering n vertices with k connected components.

Original entry on oeis.org

1, 1, 1, 5, 3, 1, 84, 23, 6, 1, 6348, 470, 65, 10, 1, 7743728, 39598, 1575, 145, 15, 1, 2414572893530, 54354104, 144403, 4095, 280, 21, 1, 56130437190053299918162, 19316801997024, 218033088, 402073, 9100, 490, 28, 1

Offset: 1

Gus Wiseman, Aug 03 2018

			Triangle begins:
        1
        1       1
        5       3       1
       84      23       6       1
     6348     470      65      10       1
  7743728   39598    1575     145      15       1

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

First column is A048143. Row sums are A006126.
Cf. A000272, A001187, A013922, A030019, A134954, A143543, A275307, A293510.
Cf. A317634, A317635, A317671, A317672, A317677.

blg={1,1,5,84,6348,7743728,2414572893530,56130437190053299918162} (*A048143*);
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Product[blg[[Length[s]]],{s,spn}],{spn,Select[sps[Range[n]],Length[#]==k&]}],{n,Length[blg]},{k,n}]

A321254 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.

Original entry on oeis.org

1, 3, 0, 6, 0, 0, 16, 1, 0, 0, 37, 3, 0, 0, 0, 105, 18, 2, 0, 0, 0, 279, 68, 7, 0, 0, 0, 0, 817, 293, 46, 3, 0, 0, 0, 0, 2387, 1141, 228, 17, 1, 0, 0, 0, 0, 7269, 4511, 1189, 135, 9, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 01 2018

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.

			Triangle begins:
     1
     3    0
     6    0    0
    16    1    0    0
    37    3    0    0    0
   105   18    2    0    0    0
   279   68    7    0    0    0    0
   817  293   46    3    0    0    0    0
  2387 1141  228   17    1    0    0    0    0
  7269 4511 1189  135    9    0    0    0    0    0

First column is A321229. Row sums are A007718.

Cf. A007716, A317672, A321155, A321194, A321253, A321255, A321256.

A321228 Number of non-isomorphic hypertrees of weight n with singletons.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 13, 23, 49, 100, 220

Offset: 0

Gus Wiseman, Oct 31 2018

A hypertree with singletons is a connected set system (finite set of finite nonempty sets) with density -1, where the density of a set system is the sum of sizes of the parts (weight) minus the number of parts minus the number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(7) = 23 hypertrees:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}      {{1,2,3,4,5}}
                  {{2},{1,2}}  {{1,3},{2,3}}    {{1,4},{2,3,4}}
                               {{3},{1,2,3}}    {{4},{1,2,3,4}}
                               {{1},{2},{1,2}}  {{2},{1,3},{2,3}}
                                                {{2},{3},{1,2,3}}
                                                {{3},{1,3},{2,3}}
.
  {{1,2,3,4,5,6}}        {{1,2,3,4,5,6,7}}
  {{1,2,5},{3,4,5}}      {{1,2,6},{3,4,5,6}}
  {{1,5},{2,3,4,5}}      {{1,6},{2,3,4,5,6}}
  {{5},{1,2,3,4,5}}      {{6},{1,2,3,4,5,6}}
  {{1},{1,4},{2,3,4}}    {{1},{1,5},{2,3,4,5}}
  {{1,3},{2,4},{3,4}}    {{1,2},{2,5},{3,4,5}}
  {{1,4},{2,4},{3,4}}    {{1,4},{2,5},{3,4,5}}
  {{3},{1,4},{2,3,4}}    {{1,5},{2,5},{3,4,5}}
  {{3},{4},{1,2,3,4}}    {{4},{1,2,5},{3,4,5}}
  {{4},{1,4},{2,3,4}}    {{4},{1,5},{2,3,4,5}}
  {{1},{2},{1,3},{2,3}}  {{4},{5},{1,2,3,4,5}}
  {{1},{2},{3},{1,2,3}}  {{5},{1,2,5},{3,4,5}}
  {{2},{3},{1,3},{2,3}}  {{5},{1,5},{2,3,4,5}}
                         {{1},{3},{1,4},{2,3,4}}
                         {{1},{4},{1,4},{2,3,4}}
                         {{2},{1,3},{2,4},{3,4}}
                         {{2},{3},{1,4},{2,3,4}}
                         {{2},{3},{4},{1,2,3,4}}
                         {{3},{1,4},{2,4},{3,4}}
                         {{3},{4},{1,4},{2,3,4}}
                         {{4},{1,3},{2,4},{3,4}}
                         {{4},{1,4},{2,4},{3,4}}
                         {{1},{2},{3},{1,3},{2,3}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A317672, A318697, A321155, A321227, A321229.

A322133 Regular triangle read by rows where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with k vertices.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 8, 3, 1, 0, 7, 17, 12, 3, 1, 0, 11, 46, 45, 18, 4, 1, 0, 15, 94, 141, 76, 23, 4, 1, 0, 22, 212, 432, 333, 124, 30, 5, 1, 0, 30, 416, 1231, 1254, 622, 178, 37, 5, 1, 0, 42, 848, 3346, 4601, 2914, 1058, 252, 45, 6, 1

Offset: 0

Gus Wiseman, Nov 27 2018

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Triangle begins:
    1
    0    1
    0    2    1
    0    3    2    1
    0    5    8    3    1
    0    7   17   12    3    1
    0   11   46   45   18    4    1
    0   15   94  141   76   23    4    1
    0   22  212  432  333  124   30    5    1
    0   30  416 1231 1254  622  178   37    5    1
    0   42  848 3346 4601 2914 1058  252   45    6    1
Non-isomorphic representatives of the multiset partitions counted in row 4:
  {{1,1,1,1}}        {{1,1,2,2}}      {{1,2,3,3}}    {{1,2,3,4}}
  {{1},{1,1,1}}      {{1,2,2,2}}      {{1,3},{2,3}}
  {{1,1},{1,1}}      {{1},{1,2,2}}    {{3},{1,2,3}}
  {{1},{1},{1,1}}    {{1,2},{1,2}}
  {{1},{1},{1},{1}}  {{1,2},{2,2}}
                     {{2},{1,2,2}}
                     {{1},{2},{1,2}}
                     {{2},{2},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A007718.

Cf. A000664, A007716, A054923, A191646, A191970, A275421, A286520, A317672, A319719, A321155, A321254, A322114.

\\ Needs G(m,n) defined in A317533 (faster PARI).
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)={[Vecrev(p) | p <- Vec(1 + InvEulerMTS(y^n*G(n,n) + sum(k=0, n-1, y^k*(1 - y)*G(k,n))))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 15 2024

A317671 Regular triangle where T(n,k) is the number of labeled connected graphs on n + 1 vertices with k maximal blobs (2-connected components).

Original entry on oeis.org

1, 1, 3, 10, 12, 16, 238, 215, 150, 125, 11368, 7740, 4140, 2160, 1296, 1014888, 509446, 205065, 84035, 36015, 16807, 166537616, 59409952, 17393152, 5393920, 1863680, 688128, 262144, 50680432112, 12321597708, 2516756508, 563570217, 148803480, 45467730

Offset: 1

Gus Wiseman, Aug 03 2018

			Triangle begins:
        1
        1       3
       10      12      16
      238     215     150     125
    11368    7740    4140    2160    1296
  1014888  509446  205065   84035   36015   16807

Row sums are A001187. First column is A013922. Last column is A000272.

Cf. A002218, A030019, A048143, A134954, A275307, A293510, A317631, A317632, A317634, A317635, A317672, A317677.

blg={0,1,1,10,238,11368,1014888,166537616,50680432112,29107809374336} (*A013922*);
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[n^(k-1)*Product[blg[[Length[s]+1]],{s,spn}],{spn,Select[sps[Range[n-1]],Length[#]==k&]}],{n,Length[blg]},{k,n-1}]

cp's OEIS Frontend

A323818 Number of connected set-systems covering n vertices.

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A191646 Triangle read by rows: T(n,k) = number of connected multigraphs with n >= 0 edges and 1 <= k <= n+1 vertices, with no loops allowed.

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A054923 Triangle read by rows: number of connected graphs with k >= 0 edges and n nodes (1<=n<=k+1).

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A076864 Number of connected loopless multigraphs with n edges.

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A321194 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of weight n with k connected components.

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A317674 Regular triangle where T(n,k) is the number of antichains covering n vertices with k connected components.

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A321254 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.

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A321228 Number of non-isomorphic hypertrees of weight n with singletons.

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A322133 Regular triangle read by rows where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with k vertices.