cp's OEIS Frontend

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

%I A367752 #24 Mar 28 2025 07:15:26
%S A367752 1,1,4,29,256,3007,42932,721121,13982563,306967231,7527903208,
%T A367752 203977383469,6051630040496,195111205542541,6792697846367791,
%U A367752 253966747582533681,10149075292428481965,431705938073882999275,19474660918369182445456,928660364396786865580881
%N A367752 Number of shapes of labeled rooted hypertrees with n vertices.
%C A367752 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.
%C A367752 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:
%C A367752   - black vertices are unlabeled;
%C A367752   - black vertices have at least two children;
%C A367752   - the children of a black vertex are white, and are connected to it via simple edges (edges connecting only two vertices);
%C A367752   - the children of a white vertex are connected to it via hyperedges (edges connecting strictly more than two vertices).
%F A367752 E.g.f.: series reversion of log(1+x)*exp(-exp(x)+x+1).
%e A367752 For n = 3 the a(3) = 4 solutions are:
%e A367752   - the corolla with a black root which have 3 white children,
%e A367752   - and the 3 possible labeling of the hypertree with a white root which have 2 white children connected to it via a hyperedge.
%o A367752 (SageMath) R.<t>=PowerSeriesRing(QQ);(ln(1+t)*exp(-exp(t)+t+1)).reverse().egf_to_ogf().list()[1:]
%o A367752 (PARI) my(x='x+O('x^30)); Vec(serlaplace(serreverse(log(1+x)*exp(-exp(x)+x+1)))) \\ _Michel Marcus_, Nov 30 2023
%Y A367752 Cf. A052888, A210586, A364709, A364816.
%K A367752 nonn
%O A367752 1,3
%A A367752 _Paul Laubie_, Nov 29 2023