

A097236


Number of hierarchical orderings ("societies") with at least 2 elements ("individuals") on each level for n labeled elements.


3



0, 1, 1, 10, 31, 271, 1534, 14393, 117653, 1253524, 13140557, 160679069, 2026451948, 28278471729, 413532314433, 6516434058758, 107958571213579, 1899723866781859, 35092386753388698, 682552407940860353, 13893916425860413469, 296049402365644855888
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OFFSET

1,4


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..175
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 8389.


FORMULA

E.g.f.: exp((exp(z)+1+z)/(2exp(z)+z)).
a(n) ~ exp(1/(2*(c2)) + 1/(2*(c1)^2) + 2*sqrt(n/((c2)*(c1)))  n  1) * n^(n1/4) / (sqrt(2) * (c1)^(1/4) * (c2)^(n+1/4)), where c = LambertW(1, exp(2)) = A226572 = 3.14619322062... .  Vaclav Kotesovec, Sep 08 2014


EXAMPLE

a(4) = 10. Let : denote the partition of n labeled individuals 1,2,3,4 into x=2 sets (i.e. "societies"). E.g. in 12:34 one has a single society with members 1 and 2 and a further single society with members 3 and 4. Let  denote a higher level (within a single society), e.g. in 12 the individual 2 is one level up with respect to individual 1. The order of individuals on a level is insignificant, eg. 1234 is equivalent to 2143.
For n = 4 and x = 2 one has 1234; 12:34; 13:24; 14:23; 1234; 3142; 4321; 2413; 2134; 4312; which gives 10 different hierarchical societies with at least 2 labeled individuals per level.


MAPLE

with(combstruct); SetSeqSetxL:=[T, {T=Set(S), S=Sequence(U, card>=1), U=Set(Z, card >= 2)}, labeled];
# where x is an integer 1, 2, 3, ... ; x=2 gives 2 individuals per level.
seq (count (SetSeqSetxL, size=j), j=1..20);


MATHEMATICA

terms = 22;
CoefficientList[ Exp[(Exp[z]+1+z)/(2Exp[z]+z)] + O[z]^(terms+1), z] * Range[0, terms]! // Rest (* JeanFrançois Alcover, Aug 06 2018 *)


CROSSREFS

Cf. A075729, A097237, A226572.
Sequence in context: A267564 A289201 A042849 * A280202 A061485 A136335
Adjacent sequences: A097233 A097234 A097235 * A097237 A097238 A097239


KEYWORD

nonn


AUTHOR

Thomas Wieder, Aug 02 2004


STATUS

approved



