A104525 The number of hierarchical orderings among the parts of the integer partitions of the integer n.
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
Keywords
Examples
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:
{*:**}, {*:*}:{*}, {*}:{**}, {*:*:*}, {*}:{*}:{*}, {**}|{*}, {*}|{*:*}, {*}|{*}|{*}, {**}:{*}, {*}:{*:*}, {*}:{*}|{*}, {***}.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..750
- N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
- Thomas Wieder, Comments on A104525
Programs
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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
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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 *)
Extensions
More terms from Alois P. Heinz, Feb 02 2009
Comments