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%I #7 Mar 31 2012 14:40:21
%S 0,0,1,4,12,32,78,183,408,886
%N The number of hierarchies with at least one subhierarchy composed of exactly 3 levels and no subhierarchy with more than 3 levels.
%H N. J. A. Sloane and Thomas Wieder, <a href="http://arXiv.org/abs/math.CO/0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 83-89.
%e Let : denote the separation between two subhierarchies, e.g. 2:3 are two subhierarchies where subhierarchy s=1 contains two elements and subhierarchy s=2 contains three elements. Let | denote the separation between two levels, e.g. 2|2|1 is a hierarchy composed of three levels with two elements on levels l=1 and l=2 and one element on level l=3. For n=5 one has a(5) = 12 hierarchies where at least one subhierarchy has exactly 3 levels (and no level l > 3 is allowed):
%e 3|1|1; 1|3|1; 1|1|3; 2|2|1; 2|1|2; 1|2|2; 1|1|1:2; 1|1|1:1:1; 1|1|1:1|1;
%e 2|1|1:1; 1|2|1:1; 1|1|2:1.
%Y Cf. A034691, A097237, A097391, A000041.
%K nonn
%O 1,4
%A _Thomas Wieder_, Aug 13 2004
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