OFFSET
1,2
COMMENTS
A hypertree is a connected hypergraph of density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices. A hypergraph is k-uniform if its edges all have size k. The span of a hypertree is the union of its edges.
LINKS
Alois P. Heinz, Rows n = 1..185, flattened
FORMULA
T(n, d) = n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1).
EXAMPLE
Triangle begins:
1
3 1
16 1
125 15 1
1296 1
16807 735 140 1
262144 1
4782969 76545 1890 1
100000000 112000 1
2357947691 13835745 33264 1
The a(4,2) = 15 hypertrees:
{{1,4,5},{2,3,5}}
{{1,4,5},{2,3,4}}
{{1,3,5},{2,4,5}}
{{1,3,5},{2,3,4}}
{{1,3,4},{2,4,5}}
{{1,3,4},{2,3,5}}
{{1,2,5},{3,4,5}}
{{1,2,5},{2,3,4}}
{{1,2,5},{1,3,4}}
{{1,2,4},{3,4,5}}
{{1,2,4},{2,3,5}}
{{1,2,4},{1,3,5}}
{{1,2,3},{3,4,5}}
{{1,2,3},{2,4,5}}
{{1,2,3},{1,4,5}}
MAPLE
T:= n-> seq(n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1), d=numtheory[divisors](n)):
seq(T(n), n=1..20); # Alois P. Heinz, Aug 21 2019
MATHEMATICA
Table[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1), {n, 10}, {d, Divisors[n]}]
CROSSREFS
KEYWORD
AUTHOR
Gus Wiseman, Jul 03 2019
STATUS
approved