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A309865 Number T(n,k) of k-uniform hypergraphs on n unlabeled nodes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A309865 #26 Aug 21 2019 17:37:16
%S A309865 2,2,2,2,3,2,2,4,4,2,2,5,11,5,2,2,6,34,34,6,2,2,7,156,2136,156,7,2,2,
%T A309865 8,1044,7013320,7013320,1044,8,2,2,9,12346,1788782616656,
%U A309865 29281354514767168,1788782616656,12346,9,2
%N A309865 Number T(n,k) of k-uniform hypergraphs on n unlabeled nodes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A309865 T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k<=n.  T(n,k) = 1 for k>n.
%C A309865 See A000088 and A000665 for more references.
%H A309865 Alois P. Heinz, <a href="/A309865/b309865.txt">Rows n = 0..14, flattened</a>
%H A309865 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 A309865 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergraph">Hypergraph</a>
%F A309865 T(n,k) = T(n,n-k) for 0 <= k <= n.
%e A309865 Triangle T(n,k) begins:
%e A309865   2;
%e A309865   2, 2;
%e A309865   2, 3,    2;
%e A309865   2, 4,    4,       2;
%e A309865   2, 5,   11,       5,       2;
%e A309865   2, 6,   34,      34,       6,    2;
%e A309865   2, 7,  156,    2136,     156,    7, 2;
%e A309865   2, 8, 1044, 7013320, 7013320, 1044, 8, 2;
%e A309865   ...
%p A309865 g:= (l, i, n)-> `if`(i=0, `if`(n=0, [[]], []), [seq(map(x->
%p A309865      [x[], j], g(l, i-1, n-j))[], j=0..min(l[i], n))]):
%p A309865 h:= (p, v)-> (q-> add((s-> add(`if`(andmap(i-> irem(k[i], p[i]
%p A309865      /igcd(t, p[i]))=0, [$1..q]), mul((m-> binomial(m, k[i]*m
%p A309865      /p[i]))(igcd(t, p[i])), i=1..q), 0), t=1..s)/s)(ilcm(seq(
%p A309865     `if`(k[i]=0, 1, p[i]), i=1..q))), k=g(p, q, v)))(nops(p)):
%p A309865 b:= (n, i, l, v)-> `if`(n=0 or i=1, 2^((p-> h(p, v))([l[], 1$n]))
%p A309865      /n!, add(b(n-i*j, i-1, [l[], i$j], v)/j!/i^j, j=0..n/i)):
%p A309865 T:= proc(n, k) option remember; `if`(k>n-k,
%p A309865       T(n, n-k), b(n$2, [], k))
%p A309865     end:
%p A309865 seq(seq(T(n, k), k=0..n), n=0..9);
%Y A309865 Columns k=0..10 give: A007395, A000027, A000088, A000665, A051240, A051249, A309860, A309861, A309862, A309863, A309864.
%Y A309865 Cf. A309858 (the same as square array).
%K A309865 nonn,tabl
%O A309865 0,1
%A A309865 _Alois P. Heinz_, Aug 20 2019