OFFSET
0,3
COMMENTS
Also the number of labeled hypergraphs spanning an initial interval of positive integers with edge-sizes summing to n. - Gus Wiseman, Dec 18 2018
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..300
P. J. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes , arXiv:math/0510155 [math.CO], 2005-2006.
M. Klazar, Extremal problems for ordered hypergraphs, arXiv:math/0305048 [math.CO], 2003.
EXAMPLE
From Gus Wiseman, Dec 18 2018: (Start)
The a(3) = 7 edge-sets:
{{1,2,3}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2,3}}
{{2},{1,3}}
{{3},{1,2}}
{{1},{2},{3}}
Inequivalent representatives of the a(4) = 28 0-1 matrices:
[1111]
.
[100][1000][010][0100][001][0010][0001][110][110][1100][101][1010][1001]
[111][0111][111][1011][111][1101][1110][101][011][0011][011][0101][0110]
.
[10][100][100][1000][100][100][1000][1000][010][010][0100][0100][0010]
[01][010][010][0100][001][001][0010][0001][001][001][0010][0001][0001]
[11][101][011][0011][110][011][0101][0110][110][101][1001][1010][1100]
.
[1000]
[0100]
[0010]
[0001]
(End)
MAPLE
b:= proc(n, i, k) b(n, i, k):=`if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j,
min(n-i*j, i-1), k)*binomial(binomial(k, i), j), j=0..n/i)))
end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..23); # Alois P. Heinz, Sep 13 2019
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k]*Binomial[Binomial[k, i], j], {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, i]*(-1)^(k-i)*Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Mar 27 2006
EXTENSIONS
a(0)=1 prepended and more terms added by Alois P. Heinz, Sep 13 2019
STATUS
approved