A104500 Number of different groupings among the hierarchical orderings of n unlabeled elements.

1, 4, 11, 35, 98, 294, 832, 2401, 6774, 19137, 53466, 148994, 412233, 1136383, 3116654, 8515706, 23172455, 62836916, 169801824, 457406173, 1228382159, 3289493887, 8784935160, 23400668297, 62179339101, 164832960183, 435978612329, 1150673925933, 3030701471118

Offset: 1

Author

Thomas Wieder, Mar 11 2005

Keywords

• nonn

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 group. For n=3 one has a(3) = 11 because
{***}, {*|*|*}, {*}{*}{*}, {*:*:*}, {*:**}, {*|**}, {*:*|*}, {*:*}{*}, {*|*}{*}, {**:*}, {*}{**}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..800
• N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Crossrefs

Cf. A034691, A104460, A034899.

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: b:= etr(n-> 2^(n-1)): a:= etr(b): seq(a(n), n=1..30); # Alois P. Heinz, Apr 21 2012

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]; b = etr[Function[{n}, 2^(n-1)]]; a = etr[b]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

Formula

Euler transform of 1, 3, 7, 18, 42, 104, 244, 585, 1373, ... = A034691.

Extensions

More terms from Alois P. Heinz, Apr 21 2012