A134955 -id:A134955 - cp's OEIS frontend

A134957 Number of hyperforests with n unlabeled vertices: analog of A134955 when edges of size 1 are allowed (with no two equal edges).

1, 2, 6, 20, 75, 310, 1422, 7094, 37877, 213610, 1256422, 7641700, 47735075, 304766742, 1981348605, 13079643892, 87480944764, 591771554768, 4042991170169, 27864757592632, 193549452132550, 1353816898675732, 9529263306483357, 67457934248821368, 480019516988969011

Offset: 0

Don Knuth, Jan 26 2008

nonn

			From _Gus Wiseman_, May 20 2018: (Start)
Non-isomorphic representatives of the a(3) = 20 hyperforests are the following:
  {}
  {{1}}
  {{1,2}}
  {{1,2,3}}
  {{1},{2}}
  {{1},{2,3}}
  {{2},{1,2}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{1},{2},{3}}
  {{1},{2},{1,2}}
  {{1},{3},{2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{3},{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}
  {{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}
  {{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
(End)

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

Cf. A030019, A035053, A048143, A054921, A134955, A134957, A134959, A144959, A304867, A304911.

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
EulerT[v_List] := With[{q = etr[v[[#]]&]}, q /@ Range[Length[v]]];
ser[v_] := Sum[v[[i]] x^(i - 1), {i, 1, Length[v]}] + O[x]^Length[v];
b[n_] := Module[{v = {1}}, For[i = 2, i <= n, i++, v = Join[{1}, EulerT[EulerT[2 v]]]]; v];
seq[n_] := Module[{u = 2 b[n]}, Join[{1}, EulerT[ser[EulerT[u]]*(1 - x*ser[u]) + O[x]^n // CoefficientList[#, x]&]]];
seq[24] (* Jean-François Alcover, Feb 10 2020, after Andrew Howroyd *)

\\ here b(n) is A318494 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
seq(n)={my(u=2*b(n)); concat([1], EulerT(Vec(Ser(EulerT(u))*(1-x*Ser(u)))))} \\ Andrew Howroyd, Aug 27 2018

Euler transform of A134959. - Gus Wiseman, May 20 2018

Terms a(7) and beyond from Andrew Howroyd, Aug 27 2018

A144959 A134955(n) - A134955(n-1). Number of hyperforests spanning n unlabeled nodes without isolated vertices.

Original entry on oeis.org

1, 0, 1, 2, 5, 11, 30, 78, 223, 658, 2026, 6429, 21015, 70233, 239360, 829224, 2912947, 10356334, 37205121, 134887153, 493000086, 1814902409, 6724595543, 25061885217, 93899071368, 353514105817, 1336822098961, 5075833932200

Offset: 0

Washington Bomfim, Sep 27 2008

nonn

a(n) is the number of hyperforests with n unlabeled nodes without isolated vertices. This follows from the fact that for n>0 A134955(n-1) counts the hyperforests of order n with one or more isolated nodes.

			From _Gus Wiseman_, May 21 2018: (Start)
Non-isomorphic representatives of the a(5) = 11 hyperforests are the following:
  {{1,2,3,4,5}}
  {{1,2},{3,4,5}}
  {{1,5},{2,3,4,5}}
  {{1,2,5},{3,4,5}}
  {{1,2},{2,5},{3,4,5}}
  {{1,2},{3,5},{4,5}}
  {{1,4},{2,5},{3,4,5}}
  {{1,5},{2,5},{3,4,5}}
  {{1,3},{2,4},{3,5},{4,5}}
  {{1,4},{2,5},{3,5},{4,5}}
  {{1,5},{2,5},{3,5},{4,5}}
(End)

Cf. A030019, A035053, A048143, A054921, A134954, A134955, A134957, A144958 (unlabeled forests without isolated vertices), A144959, A304716, A304717, A304867, A304911.

etr[p_] := etr[p] = Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b];
b[0] = 0; b[n_] := b[n] = etr[etr[b]][n-1];
c[1] = 0; c[n_] := b[n] + etr[b][n] - Sum[b[k]*etr[b][n-k], {k, 0, n}];
a = etr[c];
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jul 12 2018, after Alois P. Heinz's code for A035053 *)

\\ here b is A007563 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n)={my(v=[1]);for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
seq(n)={my(u=b(n)); concat([1], EulerT(concat([0], Vec(Ser(EulerT(u))*(1-x*Ser(u))-1))))} \\ Andrew Howroyd, May 22 2018

Euler transform of b(1) = 0, b(n > 1) = A035053(n). - Gus Wiseman, May 21 2018

A144977 a(n) = A134955(n) - A134955(n-2).

Original entry on oeis.org

1, 1, 3, 7, 16, 41, 108, 301, 881, 2684, 8455, 27444, 91248, 309593, 1068584, 3742171, 13269281, 47561455, 172092274, 627887239, 2307902495, 8539497952, 31786480760, 118960956585, 447413177185, 1690336204778, 6412656031161

Offset: 1

Washington Bomfim, Sep 28 2008

nonn

a(n) is the number of hyperforests with n unlabeled nodes without trees of order 2. This follows from the fact that for n >= 2 A134955(n-2) counts the hyperforests of order n with one or more trees of order 2.
The unique hyperforest (without loops) of order 1 is an isolated vertex, so a(1) = 1.
For n >= 2, a(n) - a(n-1) counts hyperforests of order n with components of order >= 3.

			a(3) = 3 since the only options are 2 hypertrees of order 3, or the forest composed by 3 isolated nodes.

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A134955, A035053 (hypertrees).

\\ here b(n) is A007563 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
seq(n)={my(u=b(n)); my(v=Vec(Ser(EulerT(u))*(1-x*Ser(u)))); EulerT(vector(#v, n, if(n<>2, v[n], 0)))} \\ Andrew Howroyd, Aug 27 2018

A134954 Number of "hyperforests" on n labeled nodes, i.e., hypergraphs that have no cycles, assuming that each edge contains at least two vertices.

Original entry on oeis.org

1, 1, 2, 8, 55, 562, 7739, 134808, 2846764, 70720278, 2021462055, 65365925308, 2359387012261, 94042995460130, 4102781803365418, 194459091322828280, 9950303194613104995, 546698973373090998382, 32101070021048906407183, 2006125858248695722280564

Offset: 0

Don Knuth, Jan 26 2008

nonn

The part of the name "assuming that each edge contains at least two vertices" is ambiguous. It may mean that not all n vertices have to be covered by some edge of the hypergraph, i.e., it is not necessarily a spanning hyperforest. However it is common to represent uncovered vertices as singleton edges, as in my example. - Gus Wiseman, May 20 2018

			From _Gus Wiseman_, May 20 2018: (Start)
The a(3) = 8 labeled spanning hyperforests are the following:
{{1,2,3}}
{{1,3},{2,3}}
{{1,2},{2,3}}
{{1,2},{1,3}}
{{3},{1,2}}
{{2},{1,3}}
{{1},{2,3}}
{{1},{2},{3}}
(End)

D. E. Knuth: The Art of Computer Programming, Volume 4, Generating All Combinations and Partitions Fascicle 3, Section 7.2.1.4. Generating all partitions. Page 38, Algorithm H. - Washington Bomfim, Sep 25 2008

Alois P. Heinz, Table of n, a(n) for n = 0..370
N. J. A. Sloane, Transforms

Cf. A030019, A035053, A048143, A054921, A134955, A134956, A134957, A144959, A242817, A304716, A304717, A304867, A304911, A304912.

b:= proc(n) option remember; add(Stirling2(n-1,i) *n^(i-1), i=0..n-1) end: B:= proc(n) x-> add(b(k) *x^k/k!, k=0..n) end: a:= n-> coeff(series(exp(B(n)(x)), x, n+1), x,n) *n!: seq(a(n), n=0..30);  # Alois P. Heinz, Sep 09 2008

b[n_] := b[n] = Sum[StirlingS2[n-1, i]*n^(i-1), {i, 0, n-1}]; B[n_][x_] := Sum[b[k] *x^k/k!, {k, 0, n}]; a[0]=1; a[n_] := SeriesCoefficient[ Exp[B[n][x]], {x, 0, n}] *n!; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

Exponential transform of A030019. - N. J. A. Sloane, Jan 30 2008
Binomial transform of A304911. - Gus Wiseman, May 20 2018
a(n) = Sum of n!*Product_{k=1..n} (A030019(k)/k!)^c_k / (c_k)! over all the partitions of n, c_1 + 2c_2 + ... + nc_n; c_1, c_2, ..., c_n >= 0. - Washington Bomfim, Sep 25 2008
a(n) ~ exp((n+1)/LambertW(1)) * n^(n-2) / (sqrt(1+LambertW(1)) * exp(2*n+2) * (LambertW(1))^n). - Vaclav Kotesovec, Jul 26 2014

A304867 Number of non-isomorphic hypertrees of weight n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 6, 13, 20, 41, 70, 144, 266, 545, 1072, 2210, 4491, 9388, 19529, 41286, 87361, 186657, 399927, 862584, 1866461, 4058367, 8852686, 19384258, 42570435, 93783472, 207157172, 458805044, 1018564642, 2266475432, 5053991582, 11292781891, 25280844844

Offset: 0

Gus Wiseman, May 20 2018

nonn

A hypertree E is a connected antichain of finite sets satisfying Sum_{e in E} (|e| - 1) = |U(E)| - 1. The weight of a hypertree is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices (see A035053).
From Kevin Ryde, Feb 25 2020: (Start)
a(n), except at n=1, is the number of free trees of n edges (so n+1 vertices) where any two leaves are an even distance apart.  All trees are bipartite graphs and this condition is equivalent to all leaves being in the same bipartite half.  The diameter of a tree is always between two leaves so these trees have even diameter (A000676).
The correspondence between hypertrees and these free trees is described for instance by Bacher (start of section 1.2).  In such a free tree, call a vertex "even" if it is an even distance from a leaf.  The hypertree vertices are these even vertices.  Each hyperedge is the set of vertices surrounding an odd vertex, so hypertree weight is the total number of edges in the free tree.
(End)

			Non-isomorphic representatives of the a(6) = 5 hypertrees are the following:
  {{1,2,3,4,5,6}}
  {{1,2},{1,3,4,5}}
  {{1,2,3},{1,4,5}}
  {{1,2},{1,3},{1,4}}
  {{1,2},{1,3},{2,4}}
Non-isomorphic representatives of the a(7) = 6 hypertrees are the following:
  {{1,2,3,4,5,6,7}}
  {{1,2},{1,3,4,5,6}}
  {{1,2,3},{1,4,5,6}}
  {{1,2},{1,3},{1,4,5}}
  {{1,2},{1,3},{2,4,5}}
  {{1,3},{2,4},{1,2,5}}
From _Kevin Ryde_, Feb 25 2020: (Start)
a(6) = 5 hypertrees of weight 6 and their corresponding free trees of 6 edges (7 vertices).  Each * is an "odd" vertex (odd distance to a leaf).  Each hyperedge is the set of "even" vertices surrounding an odd.
  {1,2,3,4,5,6}       3   2
                       \ /
                      4-*-1      (star 7)
                       / \
                      5   6
  .
  {1,2},{1,3,4,5}               /-3
                      2--*--1--*--4
                                \-5
  .
  {1,2,3},{1,4,5}     2-\       /-4
                         *--1--*
                      3-/       \-5
  .
  {1,2},{1,3},{1,4}    /-*--2
                      1--*--3
                       \-*--4
  .
  {1,2},{2,4},{1,3}   3--*--1--*--2--*--4   (path 7)
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.

Cf. A007716, A030019, A035053, A048143, A054921, A134955, A134957, A144959, A304911, A304912, A318601.

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
EulerT[v_List] := With[{q = etr[v[[#]]&]}, q /@ Range[Length[v]]];
ser[v_] := Sum[v[[i]] x^(i-1), {i, 1, Length[v]}] + O[x]^Length[v];
c[n_] := Module[{v = {1}}, For[i = 1, i <= Ceiling[n/2], i++, v = Join[{1}, EulerT[Join[{0}, EulerT[v]]]]]; v];
seq[n_] := Module[{u = c[n]}, x*ser[EulerT[u]]*(1 - x*ser[u]) + (1 - x)* ser[u] + x + O[x]^n // CoefficientList[#, x]&];
seq[40] (* Jean-François Alcover, Feb 08 2020, after Andrew Howroyd *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
c(n)={my(v=[1]); for(i=1, ceil(n/2), v=concat([1], EulerT(concat([0], EulerT(v))))); v}
seq(n)={my(u=c(n)); Vec(x*Ser(EulerT(u))*(1-x*Ser(u)) + (1 - x)*Ser(u) + x + O(x*x^n))} \\ Andrew Howroyd, Aug 29 2018

a(n) = Sum_{k=1..floor(n/2)} A318601(n+1-k, k). - Andrew Howroyd, Aug 29 2018

Terms a(10) and beyond from Andrew Howroyd, Aug 29 2018

A035053 Number of connected graphs on n unlabeled nodes where every block is a complete graph.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 22, 59, 165, 496, 1540, 4960, 16390, 55408, 190572, 665699, 2354932, 8424025, 30424768, 110823984, 406734060, 1502876903, 5586976572, 20884546416, 78460794158, 296124542120, 1122346648913, 4270387848473

Offset: 0

Christian G. Bower, Oct 15 1998

Equivalently, this is the number of "hypertrees" on n unlabeled nodes, i.e., connected hypergraphs that have no cycles, assuming that each edge contains at least two vertices. - Don Knuth, Jan 26 2008. See A134955 for hyperforests.

Graphs where every block is a complete graph are also called block graphs or clique tree. They can be characterized as induced-diamond-free chordal graphs. - Falk Hüffner, Jul 25 2019

			From _Gus Wiseman_, May 20 2018: (Start)
Non-isomorphic representatives of the a(5) = 9 hypertrees are the following:
  {{1,2,3,4,5}}
  {{1,5},{2,3,4,5}}
  {{1,2,5},{3,4,5}}
  {{1,2},{2,5},{3,4,5}}
  {{1,4},{2,5},{3,4,5}}
  {{1,5},{2,5},{3,4,5}}
  {{1,3},{2,4},{3,5},{4,5}}
  {{1,4},{2,5},{3,5},{4,5}}
  {{1,5},{2,5},{3,5},{4,5}}
(End)

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 71, (3.4.14).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
Robert Hellmann and Eckard Bich, A systematic formulation of the virial expansion for nonadditive interaction potentials, J. Chem. Phys. 135, 084117 (2011); doi:10.1063/1.3626524 (7 pages).
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.
Eric Weisstein's World of Mathematics, Block Graph
Eric Weisstein's World of Mathematics, Connected Graph
Wikipedia, Block graph.

Cf. A007549, A007563, A007716, A030019, A035051, A035052, A054921, A134957, A134959, A245566, A304867, A304887, A304937.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0,1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: b:= etr(B): c:= etr(b): B:= n-> if n=0 then 0 else c(n-1) fi: C:= etr(B): a:= n-> B(n)+C(n) -add(B(k)*C(n-k), k=0..n): seq(a(n), n=0..30); # Alois P. Heinz, Sep 09 2008

ClearAll[etr, b, a]; etr[p_] := etr[p] = Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; b[0]=0; b[n_] := b[n] = etr[etr[b]][n-1]; a[n_] := b[n] + etr[b][n] - Sum[b[k]*etr[b][n-k], {k, 0, n}]; Table[ a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 09 2012, after Alois P. Heinz *)

\\ here b(n) is A007563 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n)={my(v=[1]);for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
seq(n)={my(u=b(n)); Vec(1 + x*Ser(EulerT(u))*(1-x*Ser(u)))} \\ Andrew Howroyd, May 22 2018

G.f.: A(x)=1+(C(x)-1)*(1-B(x)). B: G.f. for A007563. C: G.f. for A035052.
a(n) ~ c * d^n / n^(5/2), where d = 4.189610958393826965527036454524... (see A245566), c = 0.245899549044224207821149415964395... . - Vaclav Kotesovec, Jul 26 2014
a(n) = A304937(n) - A304937(n-1) for n>1, a(n) = 1 for n<2. - Gus Wiseman, May 22 2018

A304912 Number of non-isomorphic spanning hyperforests of weight n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 18, 29, 56, 97, 186, 337, 657, 1238, 2442, 4768, 9569, 19174, 39151, 80154, 166211, 346239, 727853, 1537611, 3270710, 6989669, 15018389, 32405378, 70230238, 152772075, 333552711, 730632928, 1605459844, 3537861659, 7817447580, 17317397837

Offset: 0

Gus Wiseman, May 20 2018

nonn

A spanning hyperforest is an antichain of finite nonempty sets, which cover a set of n vertices, whose connected components are hypertrees (see A304867). The weight of a hypertree is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices (see A134957).

			The a(6) = 18 spanning hyperforests are the following:
  {{1,2,3,4,5,6}}
  {{1},{2,3,4,5,6}}
  {{1,2},{3,4,5,6}}
  {{1,5},{2,3,4,5}}
  {{1,2,3},{4,5,6}}
  {{1,2,5},{3,4,5}}
  {{1},{2},{3,4,5,6}}
  {{1},{2,3},{4,5,6}}
  {{1},{2,5},{3,4,5}}
  {{1,2},{3,4},{5,6}}
  {{1,2},{3,5},{4,5}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1},{2},{3},{4,5,6}}
  {{1},{2},{3,4},{5,6}}
  {{1},{2},{3,5},{4,5}}
  {{1},{2},{3},{4},{5,6}}
  {{1},{2},{3},{4},{5},{6}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A007716, A035053, A048143, A054921, A134954, A134955, A134957, A144959, A286520, A293993, A293994, A304911, A304867.

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
EulerT[v_List] := With[{q = etr[v[[#]]&]}, q /@ Range[Length[v]]];
ser[v_] := Sum[v[[i]] x^(i - 1), {i, 1, Length[v]}] + O[x]^Length[v];
c[n_] := Module[{v = {1}}, For[i = 1, i <= Ceiling[n/2], i++, v = Join[{1}, EulerT[Join[{0}, EulerT[v]]]]]; v];
seq[n_] := Module[{u = c[n]}, x*ser[EulerT[u]]*(1 - x*ser[u]) + (1 - x)* ser[u] + x + O[x]^n // CoefficientList[#, x]& // Rest // EulerT // Prepend[#, 1]&];
seq[36] (* Jean-François Alcover, Feb 09 2020, after Andrew Howroyd *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
c(n)={my(v=[1]); for(i=2, ceil(n/2), v=concat([1], EulerT(concat([0], EulerT(v))))); v}
seq(n)={my(u=c(n)); concat([1], EulerT(Vec(x*Ser(EulerT(u))*(1-x*Ser(u)) + (1 - x)*(Ser(u) - 1)+ O(x*x^n))))} \\ Andrew Howroyd, Aug 29 2018

Euler transform of A304867.

Terms a(10) and beyond from Andrew Howroyd, Aug 29 2018

A304911 Number of labeled hyperforests spanning n vertices without singleton edges.

Original entry on oeis.org

1, 0, 1, 4, 32, 351, 5057, 90756, 1956971, 49366904, 1427680932, 46590895869, 1694163054597, 67938488277050, 2978980898086377, 141801848209013050, 7282651452378019772, 401410357608479625207, 23635996827115264290005

Offset: 0

Gus Wiseman, May 20 2018

nonn

			The a(3) = 4 hyperforests are {{1,2,3}}, {{1,3},{2,3}}, {{1,2},{2,3}}, {{1,2},{1,3}}.

Cf. A006126, A030019, A035053, A048143, A054921, A134954, A134955, A134957, A144959, A293993, A293994, A304716, A304717, A304867.

E.g.f.: exp(A030019(x) - x - 1) where A030019(x) is the e.g.f. of A030019.

A134959 Number of spanning hypertrees with n unlabeled vertices: analog of A035053 when edges of size 1 are allowed (with no two equal edges).

Original entry on oeis.org

1, 2, 3, 10, 35, 150, 707, 3700, 20470, 119260, 719341, 4466316, 28367118, 183620874, 1207563011, 8049914664, 54295152117, 369981325578, 2544017965638, 17633790542978, 123108792874528, 865045359778662, 6114040341515978, 43443726772579152, 310195170229429300

Offset: 0

Don Knuth, Jan 26 2008

nonn

			From _Gus Wiseman_, May 20 2018: (Start)
Non-isomorphic representatives of the a(3) = 10 hypertrees are the following:
  {{1,2,3}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{3},{1,3},{2,3}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}
(End)

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

Cf. A030019, A035053, A048143, A054921, A134955, A134957, A144959, A304867, A304911, A318494.

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
EulerT[v_List] := With[{q = etr[v[[#]] &]}, q /@ Range[Length[v]]];
ser[v_] := Sum[v[[i]] x^(i - 1), {i, 1, Length[v]}] + O[x]^Length[v];
b[n_] := Module[{v = {1}}, For[i = 2, i <= n, i++, v = Join[{1}, EulerT[EulerT[2 v]]]]; v];
seq[n_] := Module[{u = 2 b[n]}, 1 + x*ser[EulerT[u]]*(1 - x*ser[u]) + O[x]^n // CoefficientList[#, x]&];
seq[25] (* Jean-François Alcover, Feb 10 2020, after Andrew Howroyd *)

\\ here b(n) is A318494 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
seq(n)={my(u=2*b(n)); Vec(1 + x*Ser(EulerT(u))*(1-x*Ser(u)))} \\ Andrew Howroyd, Aug 27 2018

Inverse Euler transform of A134957. - Gus Wiseman, May 20 2018

Terms a(7) and beyond from Andrew Howroyd, Aug 27 2018

A134956 Number of hyperforests with n labeled vertices: analog of A134954 when edges of size 1 are allowed (with no two equal edges).

Original entry on oeis.org

1, 2, 8, 64, 880, 17984, 495296, 17255424, 728771584, 36208782336, 2069977144320, 133869415030784, 9664049202221056, 770400218809384960, 67219977066339008512, 6372035504466437079040, 652103070162164448952320, 71656927837957783339925504

Offset: 0

Don Knuth, Jan 26 2008

nonn

			From _Gus Wiseman_, May 21 2018: (Start)
The a(2) = 8 hyperforests are the following:
  {{1},{2},{1,2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1,2}}
  {{1},{2}}
  {{1}}
  {{2}}
  {}
(End)

D. E. Knuth: The Art of Computer Programming, Volume 4, Generating All Combinations and Partitions Fascicle 3, Section 7.2.1.4. Generating all partitions. Page 38, Algorithm H. - Washington Bomfim, Sep 25 2008

Alois P. Heinz, Table of n, a(n) for n = 0..335

Cf. A134958. - Washington Bomfim, Sep 25 2008

Cf. A030019, A035053, A048143, A054921, A134954, A134955, A134957, A144959, A304716, A304717, A304867, A304911.

with(combinat): p:= proc(n) option remember; add(stirling2(n-1, i) *n^(i-1), i=0..n-1) end: g:= proc(n) option remember; p(n) +add(binomial(n-1, k-1) *p(k) *g(n-k), k=1..n-1) end: a:= n-> `if`(n=0, 1, 2^n * g(n)): seq(a(n), n=0..30); # Alois P. Heinz, Oct 07 2008

p[n_] := p[n] = Sum[ StirlingS2[n-1, i]*n^(i-1), {i, 0, n-1}]; g[n_] := g[n] = p[n] + Sum[Binomial[n-1, k-1]*p[k]*g[n-k], {k, 1, n-1}]; a[n_] := If[n == 0, 1, 2^n* g[n]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

Equals 2^n*A134954(n).

a(n) = Sum of n!prod_{k=1}^n\{ frac{ A134958(k)^{c_k} }{ k!^{c_k} c_k! } } over all the partitions of n, c_1 + 2c_2 + ... + nc_n; c_1, c_2, ..., c_n >= 0. - Washington Bomfim, Sep 25 2008

cp's OEIS Frontend

A134957 Number of hyperforests with n unlabeled vertices: analog of A134955 when edges of size 1 are allowed (with no two equal edges).

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A144959 A134955(n) - A134955(n-1). Number of hyperforests spanning n unlabeled nodes without isolated vertices.

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A144977 a(n) = A134955(n) - A134955(n-2).

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A134954 Number of "hyperforests" on n labeled nodes, i.e., hypergraphs that have no cycles, assuming that each edge contains at least two vertices.

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A304867 Number of non-isomorphic hypertrees of weight n.

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A035053 Number of connected graphs on n unlabeled nodes where every block is a complete graph.

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A304912 Number of non-isomorphic spanning hyperforests of weight n.

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