

A097237


Number of hierarchical orderings ("societies") of n labeled elements ("individuals") with at least two occupied levels.


6



0, 2, 12, 86, 780, 8462, 106092, 1507046, 23905740, 418581662, 8014481772, 166501716086, 3728936827980, 89530481995502, 2293539506425452, 62429371709206406, 1799021068567370700, 54707449240102350782, 1750530594833378049132, 58787407236482804618006
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OFFSET

1,2


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..150
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 8389.


FORMULA

E.g.f.: exp((exp(z)^22*exp(z)+1)/(2+exp(z))).
a(n) ~ exp(sqrt(2*n/log(2)) + 1/(4*log(2))  n  7/4) * n^(n1/4) / (2^(3/4) * log(2)^(n+1/4)).  Vaclav Kotesovec, Sep 13 2014


EXAMPLE

a(3) = 12. 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 = 3 and x = 2 one has 123; 231; 132; 123; 213; 312; 123; 231; 312; 132; 321; 213; which gives 12 different societies with at least 2 occupied levels.


MAPLE

with(combstruct); SetSeq2SetL:=[T, {T=Set(S), S=Sequence(U, card>=2), U=Set(Z, card >= 1)}, labeled];
# where x is an integer 1, 2, 3, ... ; x=2 gives 2 levels per society.
seq (count (SetSeq2SetL, size=j), j=1..12);


MATHEMATICA

Rest[CoefficientList[Series[E^((2*E^xE^(2*x)1) / (E^x2)), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Sep 13 2014 *)


CROSSREFS

Cf. A075729, A097236.
Sequence in context: A226238 A179495 A208977 * A055531 A305209 A290568
Adjacent sequences: A097234 A097235 A097236 * A097238 A097239 A097240


KEYWORD

nonn


AUTHOR

Thomas Wieder, Aug 02 2004


STATUS

approved



