OFFSET
0,4
COMMENTS
The density of a hypergraph is the sum of sizes of its edges minus the number of edges minus the number of vertices. A hypertree is a connected hypergraph of density -1. A hypergraph is uniform if its edges all have the same size. The span of a hypergraph is the union of its edges.
LINKS
Robert Israel, Table of n, a(n) for n = 0..387
FORMULA
a(n + 1) = Sum_{d|n} n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1).
a(p prime) = 1 + (p + 1)^(p - 1).
EXAMPLE
Non-isomorphic representatives of the 5 unlabeled uniform hypertrees on 5 vertices and their multiplicities in the labeled case, which add up to a(5) = 141:
5 X {{1,5},{2,5},{3,5},{4,5}}
60 X {{1,4},{2,5},{3,5},{4,5}}
60 X {{1,3},{2,4},{3,5},{4,5}}
15 X {{1,2,5},{3,4,5}}
1 X {{1,2,3,4,5}}
MAPLE
f:= proc(n) local d; add((n-1)!/(d! * ((n-1)/d)!) * (n/d)^((n-1)/d - 1), d = numtheory:-divisors(n-1)); end proc:
f(0):= 1: f(1):= 1:
map(f, [$0..25]); # Robert Israel, Jan 10 2019
MATHEMATICA
Table[Sum[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1), {d, Divisors[n]}], {n, 10}]
PROG
(PARI) a(n) = if (n<2, 1, n--; sumdiv(n, d, n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1))); \\ Michel Marcus, Jan 10 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 09 2019
STATUS
approved