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

Author

Don Knuth, Jan 26 2008

Keywords

• nonn

Examples

			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)
		

Links

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

Crossrefs

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

Programs

Mathematica

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 *)

PARI

\\ 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

Formula

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

Extensions

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