cp's OEIS Frontend

A104525 The number of hierarchical orderings among the parts of the integer partitions of the integer n.

%I A104525 #21 Dec 10 2016 15:05:58
%S A104525 1,4,12,40,123,395,1227,3851,11944,37032,114144,351040,1075316,
%T A104525 3285398,10007731,30409157,92169561,278738219,841132013,2533138770,
%U A104525 7614144053,22845435104,68427663680,204623945617,610951554377,1821438443615,5422608839874,16121857331124
%N A104525 The number of hierarchical orderings among the parts of the integer partitions of the integer n.
%C A104525 Euler transform of A055887 = number of ordered partitions of partitions.
%H A104525 Alois P. Heinz, <a href="/A104525/b104525.txt">Table of n, a(n) for n = 1..750</a>
%H A104525 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.
%H A104525 Thomas Wieder, <a href="/A104525/a104525.txt">Comments on A104525</a>
%e A104525 Let * denote an element, let : denote separator among different levels within a hierarchy, let | denote a separator between different hierarchies. Furthermore, the braces {} indicate a frame. For n=3 one has a(3) = 12 because:
%e A104525 {*:**}, {*:*}:{*}, {*}:{**}, {*:*:*}, {*}:{*}:{*}, {**}|{*}, {*}|{*:*}, {*}|{*}|{*}, {**}:{*}, {*}:{*:*}, {*}:{*}|{*}, {***}.
%p A104525 We can use combstruct to actually construct the structures A104525(n). %1 := Sequence(Set(Set(Z))).
%p A104525 with(combinat): with (numtheory): b:= proc(n) local k; option remember; `if`(n=0, 1, add (numbpart(k) * b(n-k), k=1..n)) end: a:= proc(n) option remember; `if` (n=0, 1, add (add (d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n) end: seq (a(n), n=1..30); # _Alois P. Heinz_, Feb 02 2009
%t A104525 max = 30; A055887 = CoefficientList[1/(2 - 1/QPochhammer[x, x]) + O[x]^(max + 1), x] ; s = 1/Product[(1 - x^n)^A055887[[n + 1]], {n, 1, max}] + O[x]^max; CoefficientList[s, x] // Rest (* _Jean-François Alcover_, Jan 10 2016 *)
%Y A104525 Cf. A034691, A034899, A055887, A104460, A104500, A109186.
%K A104525 nonn
%O A104525 1,2
%A A104525 _Thomas Wieder_, Mar 12 2005. Definition revised Mar 28 2009
%E A104525 More terms from _Alois P. Heinz_, Feb 02 2009