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A309868 Number of uniform hypergraphs on n unlabeled nodes with at least one (possibly empty) hyperedge.

%I A309868 #16 Aug 22 2019 20:38:33
%S A309868 1,2,4,8,20,78,2459,14028740,29284932080025185,
%T A309868 468863491068204454517854447175206,
%U A309868 1994324729204021501147398087008429477142243091610827370319897909501551
%N A309868 Number of uniform hypergraphs on n unlabeled nodes with at least one (possibly empty) hyperedge.
%C A309868 A hypergraph is called uniform if all hyperedges have the same cardinality.
%H A309868 Alois P. Heinz, <a href="/A309868/b309868.txt">Table of n, a(n) for n = 0..14</a>
%H A309868 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 A309868 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergraph">Hypergraph</a>
%F A309868 a(n) = Sum_{k=0..n} (A309865(n,k) - 1).
%e A309868 Non-isomorphic representatives of the a(3) = 8 uniform hypergraphs on 3 unlabeled nodes with at least one hyperedge: {{}}, {1}, {1,2}, {1,2,3}, {12}, {12,13}, {12,13,23}, {123}.
%p A309868 g:= (l, i, n)-> `if`(i=0, `if`(n=0, [[]], []), [seq(map(x->
%p A309868      [x[], j], g(l, i-1, n-j))[], j=0..min(l[i], n))]):
%p A309868 h:= (p, v)-> (q-> add((s-> add(`if`(andmap(i-> irem(k[i], p[i]
%p A309868      /igcd(t, p[i]))=0, [$1..q]), mul((m-> binomial(m, k[i]*m
%p A309868      /p[i]))(igcd(t, p[i])), i=1..q), 0), t=1..s)/s)(ilcm(seq(
%p A309868     `if`(k[i]=0, 1, p[i]), i=1..q))), k=g(p, q, v)))(nops(p)):
%p A309868 b:= (n, i, l, v)-> `if`(n=0 or i=1, 2^((p-> h(p, v))([l[], 1$n]))
%p A309868      /n!, add(b(n-i*j, i-1, [l[], i$j], v)/j!/i^j, j=0..n/i)):
%p A309868 T:= proc(n, k) T(n, k):=`if`(k>n-k, T(n, n-k), b(n$2, [], k)) end:
%p A309868 a:= n-> add(T(n, k)-1, k=0..n):
%p A309868 seq(a(n), n=0..10);
%Y A309868 Row sums of A309876.
%Y A309868 Cf. A309865.
%K A309868 nonn
%O A309868 0,2
%A A309868 _Peter Dolland_ and _Alois P. Heinz_, Aug 20 2019