A097236 -id:A097236 - cp's OEIS frontend

A109509 Number of hierarchical orderings with at least 2 elements on each level for n unlabeled elements. Unlabeled analog of A097236.

1, 0, 1, 1, 3, 4, 9, 14, 28, 47, 88, 152, 279, 486, 876, 1539, 2744, 4824, 8551, 15023, 26503, 46509, 81747, 143210, 251007, 438915, 767403, 1339487, 2336955, 4071906, 7090589, 12333894, 21440241, 37235775, 64624267, 112067176, 194209732, 336313393, 582019000

Offset: 0

Thomas Wieder, Jun 30 2005

nonn

A109509 is the Euler transform of the right-shifted Fibonacci numbers A000045.

			Let * denote an unlabeled element.
Let | denote a delimiter between two hierarchies. E.g., for n=3 we have in **|* two hierarchies (each with one level only).
Let : denote a higher level (within a single hierarchy). E.g., for n=6 we have in ***:**:* a single hierarchy distributed over three levels.
Then a(5) = 4 because we have *****, ***:**, **:***, **|***.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Cf. A000045, A075729, A097236, A166861.

SeqSetSetxU := [T, {T=Set(S),S=Sequence(U,card>=1),U=Set(Z,card>=2)},unlabeled]; seq(count(SeqSetSetxU,size=j),j=1..25); # where x is an integer 1, 2, 3,... # x=2 gives 2 individuals per level.

CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k-1], {k, 1, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 06 2015 *)

ET(v)=Vec(prod(k=1,#v,1/(1-x^k+x*O(x^#v))^v[k]))
ET(vector(40,n,fibonacci(n-1)))

a(n) ~ phi^(n-1/4) / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(phi/10 - 1/2 + 2*5^(-1/4)*sqrt(n/phi) + s), where s = Sum_{k>=2} 1/((phi^(2*k) - phi^k - 1)*k) = 0.189744799982532613329750744326543900883761701983311537716143669... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 06 2015

Edited with more terms from Franklin T. Adams-Watters, Oct 21 2009

A097237 Number of hierarchical orderings ("societies") of n labeled elements ("individuals") with at least two occupied levels.

Original entry on oeis.org

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

Offset: 1

Thomas Wieder, Aug 02 2004

nonn

			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 1|2 the individual 2 is one level up with respect to individual 1. The order of individuals on a level is insignificant, e.g., 12|34 is equivalent to 21|43. For n = 3 and x = 2 one has 12|3; 23|1; 13|2; 1|23; 2|13; 3|12; 1|2|3; 2|3|1; 3|1|2; 1|3|2; 3|2|1; 2|1|3; which gives 12 different societies with at least 2 occupied levels.

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), 83-89.

Cf. A075729, A097236.

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);

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

E.g.f.: exp(-(exp(z)^2-2*exp(z)+1)/(-2+exp(z))).

a(n) ~ exp(sqrt(2*n/log(2)) + 1/(4*log(2)) - n - 7/4) * n^(n-1/4) / (2^(3/4) * log(2)^(n+1/4)). - Vaclav Kotesovec, Sep 13 2014

A332255 E.g.f.: 1 / (2 - 1 / (2 + x - exp(x))).

Original entry on oeis.org

1, 0, 1, 1, 13, 41, 461, 2745, 32397, 288937, 3794605, 44758649, 665371565, 9660560937, 162652002189, 2782536864697, 52737562595917, 1033546861769513, 21867683869860845, 481630083492884601, 11277805333488014445, 275314710164399079337, 7077059249870048306125

Offset: 0

Ilya Gutkovskiy, Feb 08 2020

nonn

Cf. A000670, A032032, A050351, A097236.

nmax = 22; CoefficientList[Series[1/(2 - 1/(2 + x - Exp[x])), {x, 0, nmax}], x] Range[0, nmax]!

seq(n)={Vec(serlaplace(1/(2 - 1 / (2 + x - exp(x + O(x*x^n))))))} \\ Andrew Howroyd, Feb 08 2020

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A032032(k) * a(n-k).

a(n) ~ n! * 2^(n-1) / ((c-1) * (2*c-3)^(n+1)), where c = -LambertW(-1, -exp(-3/2)) = 2.3576766739458990584... - Vaclav Kotesovec, Feb 08 2020

cp's OEIS Frontend

A109509 Number of hierarchical orderings with at least 2 elements on each level for n unlabeled elements. Unlabeled analog of A097236.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A097237 Number of hierarchical orderings ("societies") of n labeled elements ("individuals") with at least two occupied levels.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A332255 E.g.f.: 1 / (2 - 1 / (2 + x - exp(x))).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula