A302129 Number of unlabeled uniform connected hypergraphs of weight n.

1, 1, 1, 1, 2, 1, 6, 1, 9, 10, 17, 1, 108, 1, 86, 401, 482, 1, 4469, 1, 8435, 47959, 8082, 1, 1007342, 52414, 112835, 15338453, 11899367, 1, 362657533, 1, 977129970, 9349593479, 35787684, 1771297657, 390347162497, 1, 779945988, 9360467497257, 16838238535445

Offset: 0

Author

Gus Wiseman, Jun 20 2018

Keywords

• nonn

Comments

A hypergraph is uniform if all edges have the same size. The weight of a hypergraph is the sum of cardinalities of the edges. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(8) = 9 uniform connected hypergraphs:
  {{1,2,3,4,5,6,7,8}}
  {{1,2,3,7}, {4,5,6,7}}
  {{1,2,5,6}, {3,4,5,6}}
  {{1,3,4,5}, {2,3,4,5}}
  {{1,2}, {1,3}, {2,4}, {3,4}}
  {{1,3}, {2,4}, {3,5}, {4,5}}
  {{1,4}, {2,3}, {2,4}, {3,4}}
  {{1,4}, {2,5}, {3,5}, {4,5}}
  {{1,5}, {2,5}, {3,5}, {4,5}}
		

Links

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

Crossrefs

Cf. A000005, A007716, A038041, A038041, A048143, A299353, A301481, A301920, A301924, A306019, A306021, A331508.

Programs

PARI

\\ See A331508 for T(n, k).
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
a(n) = {if(n==0, 1, sumdiv(n, d, if(d==1 || d==n, d==1, InvEulerT(vector(d, i, T(n/d, i)))[d] )))} \\ Andrew Howroyd, Jan 16 2024

Formula

a(p) = 1 for prime p. - Andrew Howroyd, Jan 16 2024

Extensions

a(11) onwards from Andrew Howroyd, Jan 16 2024