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%I #27 Jun 08 2018 10:46:02
%S 1,0,1,3,8,18,45,102,245,565,1324,3049,7066,16199,37187,84887,193532,
%T 439600,996818,2253941,5086980,11454778,25746467,57756522,129342179,
%U 289153474,645399011,1438308839,3200671082,7112360474,15783402471,34980122720,77428353682
%N Number of hierarchical orderings ("societies") of n unlabeled elements ("individuals") with at least two occupied levels.
%C Unlabeled analog of A097237.
%C Primes in this sequence include: a(3) = 3, a(11) = 3049, a(19) = 2253941, a(22) = 25746467. Semiprimes in this sequence include: a(9) = 565 = 5 * 113, a(12) = 7066 = 2 * 3533, a(13) = 16199 = 97 * 167, a(14) = 37187 = 41 * 907, a(15) = 84887 = 11 * 7717, a(18) = 996818 = 2 * 498409, a(24) = 129342179 = 23 * 5623573, a(30) = 15783402471 = 3 * 5261134157. - _Jonathan Vos Post_, Jul 10 2005
%H Alois P. Heinz, <a href="/A110045/b110045.txt">Table of n, a(n) for n = 0..1000</a>
%H N. J. A. Sloane and Thomas Wieder, <a href="https://arxiv.org/abs/math/0307064">The Number of Hierarchical Orderings</a>, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
%F G.f.: Product_{k>=1} 1/(1 - x^k)^(2^(k-1)-1). - _Ilya Gutkovskiy_, Jun 07 2018
%F a(n) ~ 2^n * exp(sqrt(2*n) - 5/4 + c) / (sqrt(2*Pi) * 2^(3/4) * n^(3/4)), where c = Sum_{k>=2} 1/(k*(2^k-1)*(2^k-2)) = 0.0927294481510243482503144824759369647388... - _Vaclav Kotesovec_, Jun 08 2018
%e Let * denote an unlabeled element.
%e Let : denote a delimiter between two levels of a hierarchy.
%e Let | denote a delimiter between two subhierarchies.
%e a(4) = 8 because we have *:*:*:*, ***:*, **:*:*, *:*|*:*, *:***, **:**, *:**:*, *:*:**.
%p SetSeqXSetU := [S, {S=Set(U), U=Sequence(V,card>=2),V=Set(Z,card>=1)},unlabeled]; seq(count(SetSeqXSetU,size=j),j=0..30); #where x is an integer 1, 2, 3,... # x=2 gives 2 levels per society.
%t nmax = 40; CoefficientList[Series[E^Sum[x^(2*k)/(k*(1 - x^k)*(1 - 2*x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 08 2018 *)
%Y Cf. A075729, A034691, A097237.
%K nonn
%O 0,4
%A _Thomas Wieder_, Jul 09 2005
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