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.

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, 11392, 12864, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 0, 0, 0, 1120, 20160, 159656, 827092, 3251736, 10389635, 27934400, 64432160, 128980800, 225774640

Offset

0,4

Comments

A hypergraph is a T_0 hypergraph if for every two distinct nodes there exists a hyperedge containing one but not the other node.

Links

Table of n, a(n) for n=0..48.

V. Jovovic, Illustration of initial terms of A059084, A059085

Formula

T(n,m) = Sum_{i=0..n} Stirling1(n,i) * binomial(2^i,m).

T(n,m) = A181230(n,m) / m!.

From Vladeta Jovovic, May 19 2004: (Start)

T(n, m) = (1/m!)*Sum_{i=0..m} s(m, i)*fallfac(2^i, n).

E.g.f.: Sum_{n>=0} (1+x)^(2^n)*log(1+y)^n/n!. (End)

Example

Triangle begins:
   m   0   1   2   3   4   5   6   7   8        sums A059085(n)
n
0      1   1                                           2
1      1   2   1                                       4
1      0   2   5   4   1                              12
2      0   0  12  44  67  56  28   8   1             216
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}}.

Mathematica

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 *)

Crossrefs

Cf. A059085, A059086.

Cf. A088309.

Sequence in context: A355650 A292892 A074142 * A246117 A295688 A355610

Adjacent sequences: A059081 A059082 A059083 * A059085 A059086 A059087

Keyword

easy,nonn,tabf

Author

Goran Kilibarda, Vladeta Jovovic, Dec 27 2000

Status

approved