A371830 Irregular triangle read by rows: T(n,k) is the number of unlabeled n-vertex hypergraphs (or set systems) with k hyperedges (none of which is empty), 0 <= k <= 2^n-1.

1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 10, 10, 6, 3, 1, 1, 4, 13, 39, 97, 187, 290, 365, 365, 290, 187, 97, 39, 13, 4, 1, 1, 5, 23, 111, 514, 2160, 8035, 26195, 74382, 183710, 395498, 744592, 1229846, 1787148, 2289929, 2591163, 2591163, 2289929, 1787148, 1229846, 744592, 395498, 183710, 74382, 26195, 8035, 2160, 514, 111, 23, 5, 1

Offset: 0

Author

Pontus von Brömssen, Apr 07 2024

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  n\k| 0  1  2  3  4   5   6   7   8   9  10 11 12 13 14 15
  ---+-----------------------------------------------------
  0  | 1
  1  | 1  1
  2  | 1  2  2  1
  3  | 1  3  6 10 10   6   3   1
  4  | 1  4 13 39 97 187 290 365 365 290 187 97 39 13  4  1
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 0..2046
• Peter H. van der Kamp, Hypergraphs and homogeneous Lotka-Volterra systems with linear Darboux polynomials, arXiv:2411.18264 [nlin.SI], 2024. See p. 4.

Crossrefs

Cf. A000612 (row sums), A008406, A052265 (empty hyperedge allowed).

Programs

SageMath

def A371830(n,k):
    return sum(1 for G in hypergraphs.nauty(k,n,set_min_size=1))

Formula

T(n,k) = Sum_{j=0..k} (-1)^(k-j)*A052265(n,j).