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A328157 Number of n-uniform hypergraphs on 2n unlabeled nodes with at least one (possibly empty) hyperedge.

%I A328157 #9 Oct 05 2019 21:06:00
%S A328157 1,2,10,2135,29281354514767167,
%T A328157 1994324729203114587259985605157804740271034553359179870979936357974015
%N A328157 Number of n-uniform hypergraphs on 2n unlabeled nodes with at least one (possibly empty) hyperedge.
%C A328157 A hypergraph is called k-uniform if all hyperedges have the same cardinality k.
%H A328157 Alois P. Heinz, <a href="/A328157/b328157.txt">Table of n, a(n) for n = 0..7</a>
%H A328157 Jianguo Qian, <a href="https://doi.org/10.1016/j.disc.2014.03.005">Enumeration of unlabeled uniform hypergraphs</a>, Discrete Math. 326 (2014), 66--74. MR3188989.
%H A328157 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergraph">Hypergraph</a>
%F A328157 a(n) = A309876(2n,n).
%p A328157 g:= (l, i, n)-> `if`(i=0, `if`(n=0, [[]], []), [seq(map(x->
%p A328157      [x[], j], g(l, i-1, n-j))[], j=0..min(l[i], n))]):
%p A328157 h:= (p, v)-> (q-> add((s-> add(`if`(andmap(i-> irem(k[i], p[i]
%p A328157      /igcd(t, p[i]))=0, [$1..q]), mul((m-> binomial(m, k[i]*m
%p A328157      /p[i]))(igcd(t, p[i])), i=1..q), 0), t=1..s)/s)(ilcm(seq(
%p A328157     `if`(k[i]=0, 1, p[i]), i=1..q))), k=g(p, q, v)))(nops(p)):
%p A328157 b:= (n, i, l, v)-> `if`(n=0 or i=1, 2^((p-> h(p, v))([l[], 1$n]))
%p A328157      /n!, add(b(n-i*j, i-1, [l[], i$j], v)/j!/i^j, j=0..n/i)):
%p A328157 a:= n-> b(2*n$2, [], n)-1:
%p A328157 seq(a(n), n=0..5);
%Y A328157 Cf. A309876.
%K A328157 nonn
%O A328157 0,2
%A A328157 _Alois P. Heinz_, Oct 05 2019