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A320444 Number of uniform hypertrees spanning n vertices. 7
1, 1, 1, 4, 17, 141, 1297, 17683, 262145, 4861405, 100112001, 2371816701, 61917364225, 1796326510993, 56693912375297, 1947734359001551, 72059082110369793, 2863257607266475419, 121439531096594251777, 5480987217944109919765, 262144000000000000000001 (list; graph; refs; listen; history; text; internal format)
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

Row sums of A326374.

Cf. A000272, A030019, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.

Sequence in context: A156076 A041026 A072755 * A129436 A063857 A072654

Adjacent sequences:  A320441 A320442 A320443 * A320445 A320446 A320447

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 09 2019

STATUS

approved

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Last modified December 8 04:31 EST 2019. Contains 329850 sequences. (Running on oeis4.)