A306021 Number of set-systems spanning {1,...,n} in which all sets have the same size.

1, 1, 2, 6, 54, 1754, 1102746, 68715913086, 1180735735356265746734, 170141183460507906731293351306656207090, 7237005577335553223087828975127304177495735363998991435497132232365910414322

Offset

0,3

Comments

a(n) is the number of labeled uniform hypergraphs spanning n vertices. - Andrew Howroyd, Jan 16 2024

Links

Andrew Howroyd, Table of n, a(n) for n = 0..14

Formula

a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(1 - k + Sum_{d = 1..k} 2^binomial(k, d)).

Inverse binomial transform of A306020. - Andrew Howroyd, Jan 16 2024

Example

The a(3) = 6 set-systems in which all sets have the same size:
  {{1,2,3}}
  {{1}, {2}, {3}}
  {{1,2}, {1,3}}
  {{1,2}, {2,3}}
  {{1,3}, {2,3}}
  {{1,2}, {1,3}, {2,3}}

Mathematica

Table[Sum[(-1)^(n-k)*Binomial[n, k]*(1+Sum[2^Binomial[k, d]-1, {d, k}]), {k, 0, n}], {n, 12}]

Prog

(PARI) a(n) = if(n==0, 1, sum(k=0, n, sum(d=0, n, (-1)^(n-d)*binomial(n, d)*2^binomial(d, k)))) \\ Andrew Howroyd, Jan 16 2024

Crossrefs

Row sums of A299471.

The unlabeled version is A301481.

The connected version is A299353.

Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306017, A306018, A306019, A306020.

Sequence in context: A085078 A152543 A279454 * A122593 A267348 A264610

Adjacent sequences: A306018 A306019 A306020 * A306022 A306023 A306024

Keyword

nonn

Author

Gus Wiseman, Jun 17 2018

Status

approved