This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A059084 #28 Dec 12 2022 18:23:43
%S A059084 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 A059084 11392,12864,11440,8008,4368,1820,560,120,16,1,0,0,0,1120,20160,
%U A059084 159656,827092,3251736,10389635,27934400,64432160,128980800,225774640
%N A059084 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 A059084 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 A059084 V. Jovovic, <a href="/A059084/a059084.pdf">Illustration of initial terms of A059084, A059085</a>
%F A059084 T(n,m) = Sum_{i=0..n} Stirling1(n,i) * binomial(2^i,m).
%F A059084 T(n,m) = A181230(n,m) / m!.
%F A059084 From _Vladeta Jovovic_, May 19 2004: (Start)
%F A059084 T(n, m) = (1/m!)*Sum_{i=0..m} s(m, i)*fallfac(2^i, n).
%F A059084 E.g.f.: Sum_{n>=0} (1+x)^(2^n)*log(1+y)^n/n!. (End)
%e A059084 Triangle begins:
%e A059084 m 0 1 2 3 4 5 6 7 8 sums A059085(n)
%e A059084 n
%e A059084 0 1 1 2
%e A059084 1 1 2 1 4
%e A059084 1 0 2 5 4 1 12
%e A059084 2 0 0 12 44 67 56 28 8 1 216
%e A059084 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 A059084 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 A059084 Cf. A059085, A059086.
%Y A059084 Cf. A088309.
%K A059084 easy,nonn,tabf
%O A059084 0,4
%A A059084 Goran Kilibarda, _Vladeta Jovovic_, Dec 27 2000