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

1, 4, 13, 46, 154, 533, 1802, 6137, 20729, 69971, 235193, 789000, 2639004, 8807811, 29327841, 97456878, 323206002, 1069923013, 3535612108, 11664423298, 38422208659, 126374059558, 415069188175, 1361443135562, 4459861400156, 14591869576268, 47686017637926

Offset: 1

Author

Thomas Wieder, Mar 09 2005

Keywords

• nonn

Comments

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.

Examples

			For n=3 there are 13 orderings:
*|*|*; *|**; *|*:a; *|*:b; ***; **|a; *:aa; *:a:a; **|b; *:bb; *:b:b; *:a:b; *:b:a.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..600
• N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Crossrefs

Cf. A034691.

Programs

Maple

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

Mathematica

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 *)
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 *)

Formula

G.f.: 1 + Sum_{n>=1} a(n) * x^n = 1 / Product_{n>=1} (1-x^n)^(3^(n-1)).

A104460 is the Euler transform of powers of 3 [1, 3, 9, 27, 81, ...].

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