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A367752
Number of shapes of labeled rooted hypertrees with n vertices.
1
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).
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
KEYWORD
nonn
AUTHOR
Paul Laubie, Nov 29 2023
STATUS
approved