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%I #28 Dec 12 2022 18:23:43
%S 1,1,1,2,1,0,2,5,4,1,0,0,12,44,67,56,28,8,1,0,0,12,268,1411,4032,7840,
%T 11392,12864,11440,8008,4368,1820,560,120,16,1,0,0,0,1120,20160,
%U 159656,827092,3251736,10389635,27934400,64432160,128980800,225774640
%N Triangle T(n,m) of number of labeled n-node T_0-hypergraphs with m distinct hyperedges (empty hyperedge included), m=0,1,...,2^n.
%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="/A059084/a059084.pdf">Illustration of initial terms of A059084, A059085</a>
%F T(n,m) = Sum_{i=0..n} Stirling1(n,i) * binomial(2^i,m).
%F T(n,m) = A181230(n,m) / m!.
%F From _Vladeta Jovovic_, May 19 2004: (Start)
%F T(n, m) = (1/m!)*Sum_{i=0..m} s(m, i)*fallfac(2^i, n).
%F E.g.f.: Sum_{n>=0} (1+x)^(2^n)*log(1+y)^n/n!. (End)
%e Triangle begins:
%e m 0 1 2 3 4 5 6 7 8 sums A059085(n)
%e n
%e 0 1 1 2
%e 1 1 2 1 4
%e 1 0 2 5 4 1 12
%e 2 0 0 12 44 67 56 28 8 1 216
%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, m], {i, 0, n}]; Table[T[n, m], {n, 0, 5}, {m, 0, 2^n}] // Flatten (* _Jean-François Alcover_, Sep 02 2016 *)
%Y Cf. A059085, A059086.
%Y Cf. A088309.
%K easy,nonn,tabf
%O 0,4
%A Goran Kilibarda, _Vladeta Jovovic_, Dec 27 2000
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