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%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
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