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A110045 Number of hierarchical orderings ("societies") of n unlabeled elements ("individuals") with at least two occupied levels. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Unlabeled analog of A097237.

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

FORMULA

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

EXAMPLE

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 *:*:*:*, ***:*, **:*:*, *:*|*:*, *:***, **:**, *:**:*, *:*:**.

MAPLE

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.

MATHEMATICA

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 *)

CROSSREFS

Cf. A075729, A034691, A097237.

Sequence in context: A026384 A322496 A066143 * A108931 A307397 A032100

Adjacent sequences:  A110042 A110043 A110044 * A110046 A110047 A110048

KEYWORD

nonn

AUTHOR

Thomas Wieder, Jul 09 2005

STATUS

approved

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Last modified April 19 22:22 EDT 2019. Contains 322291 sequences. (Running on oeis4.)