OFFSET
0,3
COMMENTS
a(n) is the number of labeled uniform hypergraphs spanning n vertices. - Andrew Howroyd, Jan 16 2024
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..14
FORMULA
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(1 - k + Sum_{d = 1..k} 2^binomial(k, d)).
Inverse binomial transform of A306020. - Andrew Howroyd, Jan 16 2024
EXAMPLE
The a(3) = 6 set-systems in which all sets have the same size:
{{1,2,3}}
{{1}, {2}, {3}}
{{1,2}, {1,3}}
{{1,2}, {2,3}}
{{1,3}, {2,3}}
{{1,2}, {1,3}, {2,3}}
MATHEMATICA
Table[Sum[(-1)^(n-k)*Binomial[n, k]*(1+Sum[2^Binomial[k, d]-1, {d, k}]), {k, 0, n}], {n, 12}]
PROG
(PARI) a(n) = if(n==0, 1, sum(k=0, n, sum(d=0, n, (-1)^(n-d)*binomial(n, d)*2^binomial(d, k)))) \\ Andrew Howroyd, Jan 16 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 17 2018
STATUS
approved