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

1,5

Links

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

Wikipedia, Hypergraph

Formula

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

Example

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

Mathematica

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

Prog

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

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.

Sequence in context: A158390 A228836 A176419 * A102602 A156951 A357052

Adjacent sequences: A299468 A299469 A299470 * A299472 A299473 A299474

Keyword

nonn,tabl

Author

Gus Wiseman, Jun 18 2018

Status

approved