login
A104525
The number of hierarchical orderings among the parts of the integer partitions of the integer n.
2
1, 4, 12, 40, 123, 395, 1227, 3851, 11944, 37032, 114144, 351040, 1075316, 3285398, 10007731, 30409157, 92169561, 278738219, 841132013, 2533138770, 7614144053, 22845435104, 68427663680, 204623945617, 610951554377, 1821438443615, 5422608839874, 16121857331124
OFFSET
1,2
COMMENTS
Euler transform of A055887 = number of ordered partitions of partitions.
LINKS
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
Thomas Wieder, Comments on A104525
EXAMPLE
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:
{*:**}, {*:*}:{*}, {*}:{**}, {*:*:*}, {*}:{*}:{*}, {**}|{*}, {*}|{*:*}, {*}|{*}|{*}, {**}:{*}, {*}:{*:*}, {*}:{*}|{*}, {***}.
MAPLE
We can use combstruct to actually construct the structures A104525(n). %1 := Sequence(Set(Set(Z))).
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
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Wieder, Mar 12 2005. Definition revised Mar 28 2009
EXTENSIONS
More terms from Alois P. Heinz, Feb 02 2009
STATUS
approved