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 A322705 Number of k-uniform k-regular hypergraphs spanning n vertices, for some 1 <= k <= n. 2

%I

%S 1,1,1,2,5,26,472,23342

%N Number of k-uniform k-regular hypergraphs spanning n vertices, for some 1 <= k <= n.

%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 The a(3) = 2 hypergraphs:

%e {{1},{2},{3}}

%e {{1,2},{1,3},{2,3}}

%e The a(4) = 5 hypergraphs:

%e {{1},{2},{3},{4}}

%e {{1,2},{1,3},{2,4},{3,4}}

%e {{1,2},{1,4},{2,3},{3,4}}

%e {{1,3},{1,4},{2,3},{2,4}}

%e {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

%e The a(5) = 26 hypergraphs:

%e {{1},{2},{3},{4},{5}}

%e {{1,2},{1,3},{2,4},{3,5},{4,5}}

%e {{1,2},{1,3},{2,5},{3,4},{4,5}}

%e {{1,2},{1,4},{2,3},{3,5},{4,5}}

%e {{1,2},{1,4},{2,5},{3,4},{3,5}}

%e {{1,2},{1,5},{2,3},{3,4},{4,5}}

%e {{1,2},{1,5},{2,4},{3,4},{3,5}}

%e {{1,3},{1,4},{2,3},{2,5},{4,5}}

%e {{1,3},{1,4},{2,4},{2,5},{3,5}}

%e {{1,3},{1,5},{2,3},{2,4},{4,5}}

%e {{1,3},{1,5},{2,4},{2,5},{3,4}}

%e {{1,4},{1,5},{2,3},{2,4},{3,5}}

%e {{1,4},{1,5},{2,3},{2,5},{3,4}}

%e {{1,2,3},{1,2,4},{1,3,5},{2,4,5},{3,4,5}}

%e {{1,2,3},{1,2,4},{1,4,5},{2,3,5},{3,4,5}}

%e {{1,2,3},{1,2,5},{1,3,4},{2,4,5},{3,4,5}}

%e {{1,2,3},{1,2,5},{1,4,5},{2,3,4},{3,4,5}}

%e {{1,2,3},{1,3,4},{1,4,5},{2,3,5},{2,4,5}}

%e {{1,2,3},{1,3,5},{1,4,5},{2,3,4},{2,4,5}}

%e {{1,2,4},{1,2,5},{1,3,4},{2,3,5},{3,4,5}}

%e {{1,2,4},{1,2,5},{1,3,5},{2,3,4},{3,4,5}}

%e {{1,2,4},{1,3,4},{1,3,5},{2,3,5},{2,4,5}}

%e {{1,2,4},{1,3,5},{1,4,5},{2,3,4},{2,3,5}}

%e {{1,2,5},{1,3,4},{1,3,5},{2,3,4},{2,4,5}}

%e {{1,2,5},{1,3,4},{1,4,5},{2,3,4},{2,3,5}}

%e {{1,2,3,4},{1,2,3,5},{1,2,4,5},{1,3,4,5},{2,3,4,5}}

%t Table[Sum[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 of A322706.

%Y Cf. A005176, A058891, A059441, A116539, A295193, A299353, A306021, A319056, A319189, A319190, A319612, A321721, A322704.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Dec 23 2018

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Last modified June 4 07:23 EDT 2020. Contains 334822 sequences. (Running on oeis4.)