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%I #15 Sep 02 2016 04:09:37
%S 1,1,1,0,2,3,1,0,0,12,32,35,21,7,1,0,0,12,256,1155,2877,4963,6429,
%T 6435,5005,3003,1365,455,105,15,1,0,0,0,1120,19040,140616,686476,
%U 2565260,7824375,20110025,44322135,84658665,141115975,206252025,265182375
%N Triangle T(n,m) of number of labeled n-node T_0-hypergraphs with m distinct hyperedges (empty hyperedge excluded), m=0,1,...,2^n-1.
%C A hypergraph is a T_0 hypergraph if for every two distinct nodes there exists a hyperedge containing one but not the other node.
%H V. Jovovic, <a href="/A059087/a059087.pdf">Illustration of initial terms of A059087, A059088</a>
%F T(n, m) = Sum_{i=0..n} s(n, i)*binomial(2^i-1, m), where s(n, i) are Stirling numbers of the first kind.
%F Also T(n, m) = (1/m!)*Sum_{i=0..m+1} s(m+1, i)*fallfac(2^(i-1), n). E.g.f: Sum((1+x)^(2^n-1)*log(1+y)^n/n!, n=0..infinity). - _Vladeta Jovovic_, May 19 2004
%e Triangle starts:
%e [1],
%e [1,1],
%e [0,2,3,1],
%e [0,0,12,32,35,21,7,1],
%e ...;
%e There are 12 labeled 3-node T_0-hypergraphs with 2 distinct hyperedges:{{3},{2}}, {{3},{2,3}}, {{2},{2,3}}, {{3},{1}}, {{3},{1,3}}, {{2},{1}}, {{2,3},{1,3}}, {{2},{1,2}}, {{2,3},{1,2}}, {{1},{1,3}}, {{1},{1,2}}, {{1,3},{1,2}}.
%t T[n_, m_] := Sum[StirlingS1[n, i] Binomial[2^i - 1, m], {i, 0, n}]; Table[T[n, m], {n, 0, 5}, {m, 0, 2^n - 1}] // Flatten (* _Jean-François Alcover_, Sep 02 2016 *)
%Y Cf. A059084, A059085, A059086, A059088, A059089.
%K easy,nonn,tabf
%O 0,5
%A Goran Kilibarda, _Vladeta Jovovic_, Dec 27 2000
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