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%I #27 Dec 10 2016 10:33:30
%S 1,0,1,1,3,4,9,14,28,47,88,152,279,486,876,1539,2744,4824,8551,15023,
%T 26503,46509,81747,143210,251007,438915,767403,1339487,2336955,
%U 4071906,7090589,12333894,21440241,37235775,64624267,112067176,194209732,336313393,582019000
%N Number of hierarchical orderings with at least 2 elements on each level for n unlabeled elements. Unlabeled analog of A097236.
%C A109509 is the Euler transform of the right-shifted Fibonacci numbers A000045.
%H Alois P. Heinz, <a href="/A109509/b109509.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1508.01796">Asymptotics of the Euler transform of Fibonacci numbers</a>, arXiv:1508.01796 [math.CO], Aug 07 2015.
%H 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 a(n) ~ phi^(n-1/4) / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(phi/10 - 1/2 + 2*5^(-1/4)*sqrt(n/phi) + s), where s = Sum_{k>=2} 1/((phi^(2*k) - phi^k - 1)*k) = 0.189744799982532613329750744326543900883761701983311537716143669... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Aug 06 2015
%e Let * denote an unlabeled element.
%e Let | denote a delimiter between two hierarchies. E.g., for n=3 we have in **|* two hierarchies (each with one level only).
%e Let : denote a higher level (within a single hierarchy). E.g., for n=6 we have in ***:**:* a single hierarchy distributed over three levels.
%e Then a(5) = 4 because we have *****, ***:**, **:***, **|***.
%p SeqSetSetxU := [T, {T=Set(S),S=Sequence(U,card>=1),U=Set(Z,card>=2)},unlabeled]; seq(count(SeqSetSetxU,size=j),j=1..25); # where x is an integer 1, 2, 3,... # x=2 gives 2 individuals per level.
%t CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k-1], {k, 1, 40}], {x, 0, 40}], x] (* _Vaclav Kotesovec_, Aug 06 2015 *)
%o (PARI) ET(v)=Vec(prod(k=1,#v,1/(1-x^k+x*O(x^#v))^v[k]))
%o ET(vector(40,n,fibonacci(n-1)))
%Y Cf. A000045, A075729, A097236, A166861.
%K nonn
%O 0,5
%A _Thomas Wieder_, Jun 30 2005
%E Edited with more terms from _Franklin T. Adams-Watters_, Oct 21 2009
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