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%I #12 Jan 11 2019 20:34:35
%S 1,0,1,0,0,1,0,0,1,1,0,0,1,5,1,0,0,1,388,16,1,0,0,1,477965,27626,42,1
%N Regular triangle read by rows where T(n,k) is the number of k-uniform hypergraphs spanning n labeled vertices where every two vertices appear together in some edge, n >= 0, 0 <= k <= n.
%e Triangle begins:
%e 1
%e 0 1
%e 0 0 1
%e 0 0 1 1
%e 0 0 1 5 1
%e 0 0 1 388 16 1
%e 0 0 1 477965 27626 42 1
%t Table[Length[Select[Subsets[If[k==0,{},Subsets[Range[n],{k}]]],And[Union@@#==Range[n],Length[Union@@(Subsets[#,{2}]&/@#)]==Binomial[n,2]]&]],{n,0,6},{k,0,n}]
%Y Row sums are A321134. Column k = 3 is A302394 without the initial terms.
%Y Cf. A058891, A116539, A301922, A306021, A319189, A320444.
%K nonn,tabl,more
%O 1,14
%A _Gus Wiseman_, Jan 10 2019
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