OFFSET
0,5
LINKS
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.
FORMULA
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
EXAMPLE
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 *****, ***:**, **:***, **|***.
MAPLE
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.
MATHEMATICA
CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k-1], {k, 1, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 06 2015 *)
PROG
(PARI) ET(v)=Vec(prod(k=1, #v, 1/(1-x^k+x*O(x^#v))^v[k]))
ET(vector(40, n, fibonacci(n-1)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Wieder, Jun 30 2005
EXTENSIONS
Edited with more terms from Franklin T. Adams-Watters, Oct 21 2009
STATUS
approved