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

1,2

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.

Formula

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

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

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 *)

Crossrefs

Cf. A034691, A104460, A034899.

Sequence in context: A297608 A034756 A029853 * A119716 A000626 A036364

Adjacent sequences: A104497 A104498 A104499 * A104501 A104502 A104503

Keyword

nonn

Author

Thomas Wieder, Mar 11 2005

Extensions

More terms from Alois P. Heinz, Apr 21 2012

Status

approved