1,2

Number of different hierarchical orderings that can be formed from n unlabeled elements: these are divided into groups and the elements in each group are then arranged in an "unlabeled preferential arrangement" or "composition" as in A000079. - Thomas Wieder (wieder.thomas(AT)t-online.de) and N. J. A. Sloane (njas(AT)research.att.com), Jun 10 2003

Thomas Wieder: The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [From Thomas Wieder (thomas.wieder(AT)t-online.de), Nov 14 2009]

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Thomas Wieder, An explicit formula for the n-th term

G.f.: 1 + Sum_{n=1..inf} a(n)*x^n = 1 / Product_{n=1..inf} (1-x^n)^(2^(n-1)).

Recurrence: a(n) = (1/n) * Sum_{m=1..n} a(n-m)*c(m) where c(m) = A083413(m).

oo := 101: mul( 1/(1-x^j)^(2^(j-1)), j=1..oo): series(%, x, oo): t1 := seriestolist(%); A034691 := n-> t1[n+1];

with(combstruct); SetSeqSetU := [T, {T=Set(S), S=Sequence(U, card >= 1), U=Set(Z, card >=1)}, unlabeled]; seq(count(SetSeqSetU, size=j), j=1..12);

Cf. A034899, A075729.

Sequence in context: A131630 A036884 A102291 this_sequence A000633 A036669 A091621

Adjacent sequences: A034688 A034689 A034690 this_sequence A034692 A034693 A034694

nonn

N. J. A. Sloane (njas(AT)research.att.com).