A322706 - OEIS

%I #5 Dec 24 2018 07:46:39

%S 1,1,0,1,1,0,1,3,1,0,1,12,12,1,0,1,70,330,70,1,0,1,465,11205,11205,

%T 465,1,0,1,3507,505505,2531200,505505,3507,1,0

%N Regular triangle read by rows where T(n,k) is the number of k-regular k-uniform hypergraphs spanning n vertices.

%C We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is k-uniform if all edges contain exactly k vertices, and k-regular if all vertices belong to exactly k edges. The span of a hypergraph is the union of its edges.

%e Triangle begins:

%e   1

%e   1       0

%e   1       1       0

%e   1       3       1       0

%e   1      12      12       1       0

%e   1      70     330      70       1       0

%e   1     465   11205   11205     465       1       0

%e   1    3507  505505 2531200  505505    3507       1       0

%e Row 4 counts the following hypergraphs:

%e   {{1}{2}{3}{4}}  {{12}{13}{24}{34}}  {{123}{124}{134}{234}}

%e                   {{12}{14}{23}{34}}

%e                   {{13}{14}{23}{24}}

%t Table[Table[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{k}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,1,n}],{n,1,6}]

%Y Row sums are A322705. Second column is A001205. Third column is A110101.

%Y Cf. A005176, A058891, A059441, A295193, A306021, A319056, A319189, A319190, A319612, A321721, A322704.

%K nonn,more,tabl

%O 1,8

%A _Gus Wiseman_, Dec 23 2018