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A110045
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Number of hierarchical orderings ("societies") of n unlabeled elements ("individuals") with at least two occupied levels.
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2
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1, 0, 1, 3, 8, 18, 45, 102, 245, 565, 1324, 3049, 7066, 16199, 37187, 84887, 193532, 439600, 996818, 2253941, 5086980, 11454778, 25746467, 57756522, 129342179, 289153474, 645399011, 1438308839, 3200671082, 7112360474, 15783402471, 34980122720, 77428353682
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: Product_{k>=1} 1/(1 - x^k)^(2^(k-1)-1). - Ilya Gutkovskiy, Jun 07 2018
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
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EXAMPLE
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Let * denote an unlabeled element.
Let : denote a delimiter between two levels of a hierarchy.
Let | denote a delimiter between two subhierarchies.
a(4) = 8 because we have *:*:*:*, ***:*, **:*:*, *:*|*:*, *:***, **:**, *:**:*, *:*:**.
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MAPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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