A367752 Number of shapes of labeled rooted hypertrees with n vertices.

1, 1, 4, 29, 256, 3007, 42932, 721121, 13982563, 306967231, 7527903208, 203977383469, 6051630040496, 195111205542541, 6792697846367791, 253966747582533681, 10149075292428481965, 431705938073882999275, 19474660918369182445456, 928660364396786865580881

Offset

1,3

Comments

The shape of a labeled rooted hypertree is a labeled rooted hypertrees where we replace all the maximal subtrees by a corolla rooted on a new unlabeled black vertex.

If we remove the black vertices that are the parent of only 1 white vertex, we obtain labeled rooted hypertrees with black and white vertices such that:

- black vertices are unlabeled;

- black vertices have at least two children;

- the children of a black vertex are white, and are connected to it via simple edges (edges connecting only two vertices);

- the children of a white vertex are connected to it via hyperedges (edges connecting strictly more than two vertices).

Links

Table of n, a(n) for n=1..20.

Formula

E.g.f.: series reversion of log(1+x)*exp(-exp(x)+x+1)).

Example

For n = 3 the a(3) = 4 solutions are:
  - the corolla with a black root which have 3 white children,
  - and the 3 possible labeling of the hypertree with a white root which have 2 white children connected to it via a hyperedge.

Prog

(SageMath) R.<t>=PowerSeriesRing(QQ); (ln(1+t)*exp(-exp(t)+t+1)).reverse().egf_to_ogf().list()[1:]
(PARI) my(x='x+O('x^30)); Vec(serlaplace(serreverse(log(1+x)*exp(-exp(x)+x+1)))) \\ Michel Marcus, Nov 30 2023

Crossrefs

Cf. A052888, A210586, A364709, A364816.

Sequence in context: A208812 A291103 A125808 * A254124 A203970 A250885

Adjacent sequences: A367749 A367750 A367751 * A367753 A367754 A367755

Keyword

nonn

Author

Paul Laubie, Nov 29 2023

Status

approved