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Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have two vertices in common.
4

%I #30 Feb 12 2022 20:30:48

%S 1,0,0,1,11,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,

%T 231,253,276,300,325,351,378,406,435,465,496,528,561,595,630,666,703,

%U 741,780,820,861,903,946,990,1035,1081,1128,1176,1225,1275,1326,1378,1431

%N Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have two vertices in common.

%H Andrew Howroyd, <a href="/A323294/b323294.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = binomial(n,2) for n >= 5. - _Gus Wiseman_, Jan 16 2019

%F Binomial transform is A289837. - _Gus Wiseman_, Jan 16 2019

%F a(n) = A000217(n-1) for n >= 5. - _Alois P. Heinz_, Jan 24 2019

%F E.g.f.: 1 - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2. - _Andrew Howroyd_, Aug 18 2019

%e The a(4) = 11 hypergraphs:

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

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

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

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

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

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

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

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

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

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

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

%t 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 Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]<=1&],Union@@#==Range[n]&]],{n,10}]

%o (PARI) seq(n)={Vec(serlaplace(1 - 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 Cf. A000217, A000665, A025035, A125791, A161680, A289837, A302374, A302394, A319540, A320395, A322451, A323292-A323299.

%K nonn

%O 0,5

%A _Gus Wiseman_, Jan 10 2019