A326374 Irregular triangle read by rows where T(n,d) for d|n is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices.

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, 61917364224, 1, 1792160394037, 3859590735, 270670400, 35135100, 720720, 1, 56693912375296, 1, 1946195068359375

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

Row sums are A320444.

Column d = 1 is A000272.

Cf. A030019, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.

Sequence in context: A160616 A362847 A168319 * A143565 A143018 A102012

Adjacent sequences: A326371 A326372 A326373 * A326375 A326376 A326377

Keyword

nonn,look,tabf

Author

Gus Wiseman, Jul 03 2019

Status

approved