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A034691
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Euler transform of powers of 2 [ 1,2,4,8,16,... ].
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18
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1, 1, 3, 7, 18, 42, 104, 244, 585, 1373, 3233, 7533, 17547, 40591, 93711, 215379, 493735, 1127979, 2570519, 5841443, 13243599, 29953851, 67604035, 152258271, 342253980, 767895424, 1719854346, 3845443858
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of different hierarchical orderings that can be formed from n unlabeled elements: these are divided into groups and the elements in each group are then arranged in an "unlabeled preferential arrangement" or "composition" as in A000079. - Thomas Wieder (wieder.thomas(AT)t-online.de) and N. J. A. Sloane (njas(AT)research.att.com), Jun 10 2003
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REFERENCES
| Thomas Wieder: The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [From Thomas Wieder (thomas.wieder(AT)t-online.de), Nov 14 2009]
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..300
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
Thomas Wieder, An explicit formula for the n-th term
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FORMULA
| G.f.: 1 / Product_{n=1..inf} (1-x^n)^(2^(n-1)).
Recurrence: a(n) = (1/n) * Sum_{m=1..n} a(n-m)*c(m) where c(m) = A083413(m).
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EXAMPLE
| 1 + x + 3*x^2 + 7*x^3 + 18*x^4 + 42*x^5 + 104*x^6 + 244*x^7 + 585*x^8 + ...
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MAPLE
| oo := 101: mul( 1/(1-x^j)^(2^(j-1)), j=1..oo): series(%, x, oo): t1 := seriestolist(%); A034691 := n-> t1[n+1];
with(combstruct); SetSeqSetU := [T, {T=Set(S), S=Sequence(U, card >= 1), U=Set(Z, card >=1)}, unlabeled]; seq(count(SetSeqSetU, size=j), j=1..12);
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MATHEMATICA
| nn = 30; b = Table[2^n, {n, 0, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Nov 21 2011 *)
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod( k=1, n, 1 / (1 - x^k + A) ^ 2^(k - 1)), n))} /* Michael Somos, Nov 21 2011 */
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CROSSREFS
| Cf. A034899, A075729.
Sequence in context: A131630 A036884 A102291 * A000633 A036669 A091621
Adjacent sequences: A034688 A034689 A034690 * A034692 A034693 A034694
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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