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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.
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%I #6 Apr 10 2024 09:27:02

%S 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,

%T 97,39,13,4,1,1,5,23,111,514,2160,8035,26195,74382,183710,395498,

%U 744592,1229846,1787148,2289929,2591163,2591163,2289929,1787148,1229846,744592,395498,183710,74382,26195,8035,2160,514,111,23,5,1

%N 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.

%H Pontus von Brömssen, <a href="/A371830/b371830.txt">Table of n, a(n) for n = 0..2046</a>

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

%e Triangle begins:

%e n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

%e ---+-----------------------------------------------------

%e 0 | 1

%e 1 | 1 1

%e 2 | 1 2 2 1

%e 3 | 1 3 6 10 10 6 3 1

%e 4 | 1 4 13 39 97 187 290 365 365 290 187 97 39 13 4 1

%o (SageMath)

%o def A371830(n,k):

%o return sum(1 for G in hypergraphs.nauty(k,n,set_min_size=1))

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

%K nonn,tabf

%O 0,5

%A _Pontus von Brömssen_, Apr 07 2024