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 A309868 Number of uniform hypergraphs on n unlabeled nodes with at least one (possibly empty) hyperedge. 2
 1, 2, 4, 8, 20, 78, 2459, 14028740, 29284932080025185, 468863491068204454517854447175206, 1994324729204021501147398087008429477142243091610827370319897909501551 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A hypergraph is called uniform if all hyperedges have the same cardinality. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..14 Jianguo Qian, Enumeration of unlabeled uniform hypergraphs, Discrete Math. 326 (2014), 66--74. MR3188989. Wikipedia, Hypergraph FORMULA a(n) = Sum_{k=0..n} (A309865(n,k) - 1). EXAMPLE 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}. MAPLE g:= (l, i, n)-> `if`(i=0, `if`(n=0, [[]], []), [seq(map(x->      [x[], j], g(l, i-1, n-j))[], j=0..min(l[i], n))]): h:= (p, v)-> (q-> add((s-> add(`if`(andmap(i-> irem(k[i], p[i]      /igcd(t, p[i]))=0, [\$1..q]), mul((m-> binomial(m, k[i]*m      /p[i]))(igcd(t, p[i])), i=1..q), 0), t=1..s)/s)(ilcm(seq(     `if`(k[i]=0, 1, p[i]), i=1..q))), k=g(p, q, v)))(nops(p)): b:= (n, i, l, v)-> `if`(n=0 or i=1, 2^((p-> h(p, v))([l[], 1\$n]))      /n!, add(b(n-i*j, i-1, [l[], i\$j], v)/j!/i^j, j=0..n/i)): T:= proc(n, k) T(n, k):=`if`(k>n-k, T(n, n-k), b(n\$2, [], k)) end: a:= n-> add(T(n, k)-1, k=0..n): seq(a(n), n=0..10); CROSSREFS Row sums of A309876. Cf. A309865. Sequence in context: A100447 A241891 A112278 * A059731 A001315 A216643 Adjacent sequences:  A309865 A309866 A309867 * A309869 A309870 A309871 KEYWORD nonn AUTHOR Peter Dolland and Alois P. Heinz, Aug 20 2019 STATUS approved

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Last modified December 15 22:02 EST 2019. Contains 330012 sequences. (Running on oeis4.)