A054921 -id:A054921 - cp's OEIS frontend

A304867 Number of non-isomorphic hypertrees of weight n.

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

A322700 Number of unlabeled graphs with loops spanning n vertices.

Original entry on oeis.org

1, 1, 4, 14, 70, 454, 4552, 74168, 2129348, 111535148, 10812483376, 1945437208224, 650378721156736, 404749938336404704, 470163239887698967104, 1022592854829028311090816, 4177826139658552046627175072, 32163829440870460348768023969632

Offset: 0

Gus Wiseman, Dec 23 2018

nonn

The span of a graph is the union of its edges. The not necessarily spanning case is A000666.

Andrew Howroyd, Table of n, a(n) for n = 0..80
Gus Wiseman, Non-isomorphic representatives of the a(4) = 70 spanning graphs with loops.

Cf. A000666, A006125, A006129, A054921, A062740, A320461, A322635, A322661.

Table[Sum[2^PermutationCycles[Ordering[Map[Sort,Select[Tuples[Range[n],2],OrderedQ]/.Rule@@@Table[{i,prm[[i]]},{i,n}],{1}]],Length],{prm,Permutations[Range[n]]}]/n!,{n,0,8}]//Differences (* Mathematica 8.0+ *)

from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A322700(n): return int(sum(Fraction(1<>1)+1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))-sum(Fraction(1<>1)+1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n-1))) if n else 1 # Chai Wah Wu, Jul 14 2024

First differences of A000666.

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

A306005 Number of non-isomorphic set-systems of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 12, 19, 51, 106, 274, 647, 1773, 4664, 13418, 38861, 118690, 370588, 1202924, 4006557, 13764760, 48517672, 175603676, 651026060, 2471150365, 9590103580, 38023295735, 153871104726, 635078474978, 2671365285303, 11444367926725, 49903627379427

Offset: 0

Gus Wiseman, Jun 16 2018

nonn

A set-system is a finite set of finite nonempty sets (edges). The weight is the sum of cardinalities of the edges. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(6) = 12 set-systems:
  {{1,2,3,4,5,6}}
  {{1,2},{3,4,5,6}}
  {{1,5},{2,3,4,5}}
  {{3,4},{1,2,3,4}}
  {{1,2,3},{4,5,6}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{3,4},{5,6}}
  {{1,2},{3,5},{4,5}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}

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

The complement is counted by A330053.

Cf. A007716, A034691, A048143, A049311, A054921, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306008.

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)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k), -k)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t),x,x^t) )); s+=permcount(q)*polcoef(exp(g - subst(g,x,x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2024

a(n) = A283877(n) - A330053(n). - Gus Wiseman, Dec 09 2019

Terms a(11) and beyond from Andrew Howroyd, Sep 01 2019

A322395 Number of labeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

Original entry on oeis.org

1, 1, 1, 4, 26, 548, 22504, 1708336, 241874928, 65285161232, 34305887955616, 35573982726480064, 73308270568902715136, 301210456065963448091072, 2471487759846321319412778624, 40526856087731237340916330352896, 1328570640536613080046570271722309632

Offset: 0

Gus Wiseman, Dec 06 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Graph Bridge
Eric Weisstein's World of Mathematics, Endpoint

Cf. A001187, A006125, A007146, A013922, A054921, A095983, A322338, A322387, A322394.

nmax = 16;
seq[n_] := Module[{v, p, q, c}, v[_] = 0; p = x*D[#, x]& @ Log[Sum[ 2^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^(n + 1)]; q = x*E^p; p -= q; For[k = 3, k <= n, k++, c = Coefficient[p, x, k]; v[k] = c*(k - 1)!; p -= c*q^k]; Join[{0}, Array[v, n]]];
A095983 = seq[nmax];
a[n_] := If[n<3, 1, n+Sum[Binomial[n, k]*A095983[[k+1]]*k^(n-k), {k, 1, n}]];
a /@ Range[0, nmax] (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

a(n) = n + Sum_{k=1..n} binomial(n,k)*A095983(k)*k^(n-k) for n >= 3. - Andrew Howroyd, Dec 07 2018

a(6)-a(16) from Andrew Howroyd, Dec 07 2018

A322338 Edge-connectivity of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 3, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 2

Offset: 1

Gus Wiseman, Dec 04 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

The edge-connectivity of an integer partition is the minimum number of parts that must be removed so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.

			2093 is the Heinz number of (9,6,4), corresponding to the multiset partition {{1,1},{1,2},{2,2}}, which can be made disconnected by removing only the part {1,2}, so a(2093) = 1.

Wikipedia, k-edge-connected graph

Cf. A001222, A003963, A013922, A054921, A056239, A095983, A112798, A302242, A304714, A304716, A305078, A305079, A322335, A322336, A322337.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[PrimeOmega[n]-Max@@PrimeOmega/@Select[Divisors[n],Length[csm[primeMS/@primeMS[#]]]!=1&],{n,100}]

A134955 Number of "hyperforests" on n unlabeled 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, 4, 9, 20, 50, 128, 351, 1009, 3035, 9464, 30479, 100712, 340072, 1169296, 4082243, 14438577, 51643698, 186530851, 679530937, 2494433346, 9219028889, 34280914106, 128179985474, 481694091291, 1818516190252, 6894350122452

Offset: 0

Don Knuth, Jan 26 2008

nonn

A hyperforest is an antichain of finite nonempty sets (edges) whose connected components are hypertrees. It is spanning if all vertices are covered by some edge. However, it is common to represent uncovered vertices as singleton edges. For example, {{1,2},{1,4}} and {{3},{1,2},{1,4}} may represent the same hyperforest, the former being free of singletons (see example 2) and the latter being spanning (see example 1). This is different from a hyperforest with singleton edges allowed, which is one whose non-singleton edges only are required to form an antichain. For example, {{1},{2},{1,3},{2,3}} is a hyperforest with singleton edges allowed. - Gus Wiseman, May 22 2018

Equivalently, number of block graphs on n nodes, that is, graphs where every block is a complete graph. These graphs 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(4) = 9 spanning hyperforests are the following:
  {{1,2,3,4}}
  {{1},{2,3,4}}
  {{1,2},{3,4}}
  {{1,4},{2,3,4}}
  {{1},{2},{3,4}}
  {{1},{2,4},{3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1},{2},{3},{4}}
Non-isomorphic representatives of the a(4) = 9 hyperforests spanning up to 4 vertices without singleton edges are the following:
  {}
  {{1,2}}
  {{1,2,3}}
  {{1,2,3,4}}
  {{1,2},{3,4}}
  {{1,3},{2,3}}
  {{1,4},{2,3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
(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..1000 (first 201 terms from T. D. Noe)
N. J. A. Sloane, Transforms
Wikipedia, Block graph

Cf. A030019, A035053 (hypertrees), A048143, A054921, A134954 (labeled case), A134955, A134957, A144959, A245566, A304716, A304717, A304867, A304911.

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): aa:= proc(n) option remember; B(n)+C(n) -add(B(k)*C(n-k), k=0..n) end: a:= etr(aa): seq(a(n), n=0..27); # Alois P. Heinz, Sep 09 2008

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 = etr[B]; c = etr[b]; B[n_] := If[n == 0, 0, c[n-1]]; CC = etr[B]; aa[n_] := aa[n] = B[n]+CC[n]-Sum[B[k]*CC[n-k], {k, 0, n}]; a = etr[aa]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz*)

\\ 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(Vec(x*Ser(EulerT(u))*(1-x*Ser(u)))))} \\ Andrew Howroyd, May 22 2018

Euler transform of A035053. - N. J. A. Sloane, Jan 30 2008
a(n) = Sum of prod_{k=1}^n\,{A035053(k) + c_k -1 /choose 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) ~ c * d^n / n^(5/2), where d = 4.189610958393826965527036454524... (see A245566), c = 0.36483930544... . - Vaclav Kotesovec, Jul 26 2014

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A304867 Number of non-isomorphic hypertrees of weight n.

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A322700 Number of unlabeled graphs with loops spanning n vertices.

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

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A306005 Number of non-isomorphic set-systems of weight n with no singletons.

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A322395 Number of labeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

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A322338 Edge-connectivity of the integer partition with Heinz number n.

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A134955 Number of "hyperforests" on n unlabeled nodes, i.e., hypergraphs that have no cycles, assuming that each edge contains at least two vertices.

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