login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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: A346397 A322496 A066143 * A108931 A307397 A032100

Adjacent sequences: A110042 A110043 A110044 * A110046 A110047 A110048

KEYWORD

nonn

AUTHOR

Thomas Wieder, Jul 09 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 1 23:44 EST 2022. Contains 358485 sequences. (Running on oeis4.)