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

%I A104500 #18 Dec 10 2016 15:05:29
%S A104500 1,4,11,35,98,294,832,2401,6774,19137,53466,148994,412233,1136383,
%T A104500 3116654,8515706,23172455,62836916,169801824,457406173,1228382159,
%U A104500 3289493887,8784935160,23400668297,62179339101,164832960183,435978612329,1150673925933,3030701471118
%N A104500 Number of different groupings among the hierarchical orderings of n unlabeled elements.
%H A104500 Alois P. Heinz, <a href="/A104500/b104500.txt">Table of n, a(n) for n = 1..800</a>
%H A104500 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.
%F A104500 Euler transform of 1, 3, 7, 18, 42, 104, 244, 585, 1373, ... = A034691.
%e A104500 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
%e A104500 {***}, {*|*|*}, {*}{*}{*}, {*:*:*}, {*:**}, {*|**}, {*:*|*}, {*:*}{*}, {*|*}{*}, {**:*}, {*}{**}.
%p A104500 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
%t A104500 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_ *)
%Y A104500 Cf. A034691, A104460, A034899.
%K A104500 nonn
%O A104500 1,2
%A A104500 _Thomas Wieder_, Mar 11 2005
%E A104500 More terms from _Alois P. Heinz_, Apr 21 2012