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

0,5

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.

Links

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

Formula

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

Example

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}}

Prog

(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

Crossrefs

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

Sequence in context: A107754 A181569 A274085 * A191093 A266303 A267354

Adjacent sequences: A302126 A302127 A302128 * A302130 A302131 A302132

Keyword

nonn

Author

Gus Wiseman, Jun 20 2018

Extensions

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

Status

approved