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Number of families of 3-subsets of an n-set that cover every element.
17

%I #29 Jan 16 2024 22:05:48

%S 1,0,0,1,11,958,1042642,34352419335,72057319189324805,

%T 19342812465316957316575404,1329227995591487745008054001085455444,

%U 46768052394574271874565344427028486133322470597757,1684996666696914425950059707959735374604894792118382485311245761903

%N Number of families of 3-subsets of an n-set that cover every element.

%C Number of simple 3-uniform hypergraphs without isolated vertices.

%H Andrew Howroyd, <a href="/A302374/b302374.txt">Table of n, a(n) for n = 0..25</a>

%F a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * 2^binomial(n-k,3).

%e For n=3, all families with at least two 3-subsets will cover every element.

%p seq(add((-1)^k * binomial(n,k) * 2^binomial(n-k,3), k = 0..n), n=0..15);

%t Array[Sum[(-1)^k*Binomial[#, k] 2^Binomial[# - k, 3], {k, 0, #}] &, 13, 0] (* _Michael De Vlieger_, Apr 07 2018 *)

%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n,k)*2^binomial(n-k,3)); \\ _Michel Marcus_, Apr 07 2018

%o (GAP) Flat(List([0..12],n->Sum([0..n],k->(-1)^k*Binomial(n,k)*2^Binomial(n-k,3)))); # _Muniru A Asiru_, Apr 07 2018

%Y Column 3 of A299471.

%Y Cf. A302394.

%K nonn,easy

%O 0,5

%A _Brendan McKay_, Apr 07 2018