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 A104500 Number of different groupings among the hierarchical orderings of n unlabeled elements. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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

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