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A323296 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have exactly one vertex in common.

%I A323296 #15 Aug 18 2019 12:50:21
%S A323296 1,0,0,1,11,10,25,406,4823,15436,72915,895180,11320441,71777498,
%T A323296 519354927,6155284240,82292879425,788821735656,7772567489083,
%U A323296 98329764933354,1400924444610675,17424772471470490,216091776292721021,3035845122991962688,46700545575567202903
%N A323296 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have exactly one vertex in common.
%C A323296 The only way to meet the requirements is to cover the vertices with zero or more disconnected 3-uniform hypergraphs with each edge having exactly two vertices in common (A323294). - _Andrew Howroyd_, Aug 18 2019
%H A323296 Andrew Howroyd, <a href="/A323296/b323296.txt">Table of n, a(n) for n = 0..200</a>
%F A323296 From _Andrew Howroyd_, Aug 18 2019: (Start)
%F A323296 Exponential transform of A323294.
%F A323296 E.g.f.: exp(-x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2). (End)
%e A323296 The a(4) = 11:
%e A323296   {{1,2,3},{1,2,4}}
%e A323296   {{1,2,3},{1,3,4}}
%e A323296   {{1,2,3},{2,3,4}}
%e A323296   {{1,2,4},{1,3,4}}
%e A323296   {{1,2,4},{2,3,4}}
%e A323296   {{1,3,4},{2,3,4}}
%e A323296   {{1,2,3},{1,2,4},{1,3,4}}
%e A323296   {{1,2,3},{1,2,4},{2,3,4}}
%e A323296   {{1,2,3},{1,3,4},{2,3,4}}
%e A323296   {{1,2,4},{1,3,4},{2,3,4}}
%e A323296   {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
%e A323296 The following are non-isomorphic representatives of the 3 unlabeled 3-uniform hypergraphs spanning 7 vertices with no two edges having exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 406.
%e A323296   210 X {{1,2,3},{4,6,7},{5,6,7}}
%e A323296   140 X {{1,2,3},{4,5,7},{4,6,7},{5,6,7}}
%e A323296    21 X {{1,6,7},{2,6,7},{3,6,7},{4,6,7},{5,6,7}}
%e A323296    35 X {{1,2,3},{4,5,6},{4,5,7},{4,6,7},{5,6,7}}
%p A323296 b:= n-> `if`(n<5, (n-2)*(2*n^2-6*n+3)/6, n/2)*(n-1):
%p A323296 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A323296       binomial(n-1, k-1)*b(k)*a(n-k), k=1..n))
%p A323296     end:
%p A323296 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 18 2019
%t A323296 stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
%t A323296 Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]==1&],Union@@#==Range[n]&]],{n,8}]
%o A323296 (PARI) seq(n)={Vec(serlaplace(exp(-x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x + O(x^(n-1)))/2)))} \\ _Andrew Howroyd_, Aug 18 2019
%Y A323296 Cf. A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323294, A323297.
%K A323296 nonn
%O A323296 0,5
%A A323296 _Gus Wiseman_, Jan 11 2019
%E A323296 a(11) from _Alois P. Heinz_, Aug 12 2019
%E A323296 Terms a(12) and beyond from _Andrew Howroyd_, Aug 18 2019