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A104460 Number of hierarchical orderings for n unlabeled elements with 2 possible classes for levels l>=2.

%I A104460 #28 Apr 12 2021 12:12:22
%S A104460 1,4,13,46,154,533,1802,6137,20729,69971,235193,789000,2639004,
%T A104460 8807811,29327841,97456878,323206002,1069923013,3535612108,
%U A104460 11664423298,38422208659,126374059558,415069188175,1361443135562,4459861400156,14591869576268,47686017637926
%N A104460 Number of hierarchical orderings for n unlabeled elements with 2 possible classes for levels l>=2.
%C A104460 Consider a hierarchical ordering of n unlabeled elements into groups as defined in A034691. In addition assume that each level l with l >= 2 can fall into one of two classes A and B. Let | denote a separator among different groups and let : denote a separator between levels. Furthermore, let * denote an unlabeled element which is written as "a" if it falls into class A and as "b" if it falls into class B. As an example with n=4 one can have *|*:ab. In this example one has two groups, where the second group has tree elements, one on level l=1 and two on level l=2. One of the two elements on l=2 belongs to class A, the other to class B.
%H A104460 Alois P. Heinz, <a href="/A104460/b104460.txt">Table of n, a(n) for n = 1..600</a>
%H A104460 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.
%F A104460 G.f.: 1 + Sum_{n>=1} a(n) * x^n = 1 / Product_{n>=1} (1-x^n)^(3^(n-1)).
%F A104460 A104460 is the Euler transform of powers of 3 [1, 3, 9, 27, 81, ...].
%F A104460 a(n) ~ exp(2*sqrt(n/3) - 1/6 + c/3) * 3^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (3^(j-1) - 1)). - _Vaclav Kotesovec_, Apr 12 2021
%e A104460 For n=3 there are 13 orderings:
%e A104460 *|*|*; *|**; *|*:a; *|*:b; ***; **|a; *:aa; *:a:a; **|b; *:bb; *:b:b; *:a:b; *:b:a.
%p A104460 etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0,1, add(add(d*p(d), d=numtheory[divisors](j)) *b(n-j), j=1..n)/n) end end: a:= etr(n-> 3^(n-1)): seq(a(n), n=1..30); # _Alois P. Heinz_, Sep 08 2008
%t A104460 etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[ Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a = etr[Function[3^(#-1)]]; Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Mar 05 2015, after _Alois P. Heinz_ *)
%t A104460 nmax = 30; Rest[CoefficientList[Series[Product[1/(1 - x^k)^(3^(k-1)), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Apr 12 2021 *)
%Y A104460 Cf. A034691.
%K A104460 nonn
%O A104460 1,2
%A A104460 _Thomas Wieder_, Mar 09 2005