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A104460
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Number of hierarchical orderings for n unlabeled elements with 2 possible classes for levels l>=2.
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4
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1, 4, 13, 46, 154, 533, 1802, 6137, 20729, 69971, 235193, 789000, 2639004, 8807811, 29327841, 97456878, 323206002, 1069923013, 3535612108, 11664423298, 38422208659, 126374059558, 415069188175, 1361443135562, 4459861400156, 14591869576268, 47686017637926
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Consider a hierarchical ordering of n unlabeled elements into groups as defined in A034691. In addition assume that each level l with l >= 2 can fall into one of two classes A and B. Let | be a separator among different groups and let : be a separator between levels. Furthermore, let * denote an unlabeled element which is written as "a" if it falls into class A and as "b" if it falls into class B. As an example with n=4 one can have *|*:ab. In this example one has two groups, where the second group has tree elements, one on level l=1 and two on level l=2. One of the two elements on l=2 belongs to class A, the other to class B.
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LINKS
| N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
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FORMULA
| G.f.: 1 + Sum_{n>=1} a(n) * x^n = 1 / Product_{n>=1} (1-x^n)^(3^(n-1)).
A104460 is the Euler transform of powers of 3 [1, 3, 9, 27, 81, ...].
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EXAMPLE
| For n=3 there are 13 orderings:
*|*|*; *|**; *|*:a; *|*:b; ***; **|a; *:aa; *:a:a; **|b; *:bb; *:b:b; *:a:b; *:b:a.
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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: a:= etr (n-> 3^(n-1)): seq (a(n), n=1..30); # Alois P. Heinz, Sep 08 2008
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CROSSREFS
| Cf. A034691.
Sequence in context: A155328 A096353 A034553 * A095128 A149433 A047154
Adjacent sequences: A104457 A104458 A104459 * A104461 A104462 A104463
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KEYWORD
| nonn
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AUTHOR
| Thomas Wieder (wieder.thomas(AT)t-online.de), Mar 09 2005
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