A299471 Regular triangle where T(n,k) is the number of labeled k-uniform hypergraphs spanning n vertices.

1, 1, 1, 1, 4, 1, 1, 41, 11, 1, 1, 768, 958, 26, 1, 1, 27449, 1042642, 32596, 57, 1, 1, 1887284, 34352419335, 34359509614, 2096731, 120, 1, 1, 252522481, 72057319189324805, 1180591620442534312297, 72057594021152435, 268434467, 247, 1, 1, 66376424160

Offset: 1

Author

Gus Wiseman, Jun 18 2018

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,     1;
  1,     4,       1;
  1,    41,      11,     1;
  1,   768,     958,    26,  1;
  1, 27449, 1042642, 32596, 57, 1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..91 (rows 1..13)
• Wikipedia, Hypergraph

Crossrefs

Columns 1..4 are A000012, A006129, A302374, A302396.

Row sums are A306021.

The unlabeled version is A301922.

The connected version is A299354.

Cf. A000005, A001315, A006126, A038041, A298422, A298426, A306017, A306018, A306019, A306020.

Programs

Mathematica

Table[Sum[(-1)^(n-d)*Binomial[n,d]*2^Binomial[d,k],{d,0,n}],{n,10},{k,n}]

PARI

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

Formula

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