A104500 -id:A104500 - cp's OEIS frontend

A034691 Euler transform of powers of 2 [1,2,4,8,16,...].

1, 1, 3, 7, 18, 42, 104, 244, 585, 1373, 3233, 7533, 17547, 40591, 93711, 215379, 493735, 1127979, 2570519, 5841443, 13243599, 29953851, 67604035, 152258271, 342253980, 767895424, 1719854346, 3845443858

Offset: 0

N. J. A. Sloane

nonn

This is the 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 and N. J. A. Sloane, Jun 10 2003
From Gus Wiseman, Mar 03 2016: (Start)
The original Sloane-Wieder definition, "To obtain a hierarchical ordering we partition the elements into unlabeled nonempty subsets and form a composition of each subset," [arXiv:math/0307064] has two possible meanings. The first possible meaning is that we should (1) choose a set partition pi of {1...n} and (2) for each block of pi choose a composition of the number of elements. In this case the correct number of such structures would evidently be counted by A004211 which differs from a(n) for n > 2.
The other possible meaning is that after we have done (1) and (2) above we (3) "forget" the choice of pi. We will have produced a collection M of multisets of compositions. The span of M (its set of distinct elements) is correctly counted by A034691 and it seems that non-isomorphic hierarchical orderings of unlabeled sets are nothing more than multisets of compositions. This discovery is due to Wieder. (End)
The asymptotic formula in the article by N. J. A. Sloane and Thomas Wieder, "The Number of Hierarchical Orderings" (Theorem 3) is incorrect (a multiplicative factor of 1.397... is missing, see my formula below). - Vaclav Kotesovec, Sep 08 2014
Number of partitions of n into 1 sort of 1, 2 sorts of 2's, 4 sorts of 3's, ..., 2^(k-1) sorts of k's, ... . - Joerg Arndt, Sep 09 2014
Also number of normal multiset partitions of weight n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Mar 03 2016

			The normal multiset partitions for a(4) = 18: {{1111},{1222},{1122},{1112},{1233},{1223},{1123},{1234},{1,111},{1,122},{1,112},{1,123},{11,11},{11,12},{12,12},{1,1,11},{1,1,12},{1,1,1,1}}

Vaclav Kotesovec, Table of n, a(n) for n = 0..3190 (first 300 terms from T. D. Noe)
Vaclav Kotesovec, Asymptotics of sequence A034691.
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, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
Thomas Wieder, An explicit formula for the n-th term.
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_, Nov 14 2009]

Cf. A034899, A075729, A247003, A004211, A104500 (Euler transform), A290222 (Multiset transform).

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);
# Alternative, uses EulerTransform from A358369:
a := EulerTransform(BinaryRecurrenceSequence(2, 0)):
seq(a(n), n = 0..27); # Peter Luschny, Nov 17 2022

nn = 30; b = Table[2^n, {n, 0, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}],  x] (* T. D. Noe, Nov 21 2011 *)
Table[SeriesCoefficient[E^(Sum[x^k/(1 - 2*x^k)/k, {k, 1, n}]), {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 08 2014 *)
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
allnmsp[0]={};allnmsp[1]={{{1}}};allnmsp[n_Integer]:=allnmsp[n]=Join[allnmsp[n-1],List/@allnorm[n],Join@@Function[ptn,Append[ptn,#]&/@Select[allnorm[n-Length[Join@@ptn]],OrderedQ[{Last[ptn],#}]&]]/@allnmsp[n-1]];
Apply[SequenceForm,Select[allnmsp[4],Length[Join@@#]===4&],{2}] (* to construct the example *)
Table[Length[Complement[allnmsp[n],allnmsp[n-1]]],{n,1,8}] (* Gus Wiseman, Mar 03 2016 *)

A034691(n,l=1+O('x^(n+1)))={polcoeff(1/prod(k=1,n,(l-'x^k)^2^(k-1)),n)} \\ Michael Somos, Nov 21 2011, edited by M. F. Hasler, Jul 24 2017

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(2, 0)
b = EulerTransform(a)
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

G.f.: 1 / Product_{n>=1} (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).
a(n) ~ c * 2^n * exp(sqrt(2*n)) / (sqrt(2*Pi) * exp(1/4) * 2^(3/4) * n^(3/4)), where c = exp( Sum_{k>=2} 1/(k*(2^k-2)) ) = 1.3976490050836502... (see A247003). - Vaclav Kotesovec, Sep 08 2014

A104525 The number of hierarchical orderings among the parts of the integer partitions of the integer n.

Original entry on oeis.org

1, 4, 12, 40, 123, 395, 1227, 3851, 11944, 37032, 114144, 351040, 1075316, 3285398, 10007731, 30409157, 92169561, 278738219, 841132013, 2533138770, 7614144053, 22845435104, 68427663680, 204623945617, 610951554377, 1821438443615, 5422608839874, 16121857331124

Offset: 1

Thomas Wieder, Mar 12 2005. Definition revised Mar 28 2009

nonn

Euler transform of A055887 = number of ordered partitions of partitions.

			Let * denote an element, let : denote separator among different levels within a hierarchy, let | denote a separator between different hierarchies. Furthermore, the braces {} indicate a frame. For n=3 one has a(3) = 12 because:
{*:**}, {*:*}:{*}, {*}:{**}, {*:*:*}, {*}:{*}:{*}, {**}|{*}, {*}|{*:*}, {*}|{*}|{*}, {**}:{*}, {*}:{*:*}, {*}:{*}|{*}, {***}.

Alois P. Heinz, Table of n, a(n) for n = 1..750
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
Thomas Wieder, Comments on A104525

Cf. A034691, A034899, A055887, A104460, A104500, A109186.

We can use combstruct to actually construct the structures A104525(n). %1 := Sequence(Set(Set(Z))).
with(combinat): with (numtheory): b:= proc(n) local k; option remember; `if`(n=0, 1, add (numbpart(k) * b(n-k), k=1..n)) end: a:= proc(n) option remember; `if` (n=0, 1, add (add (d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n) end: seq (a(n), n=1..30); # Alois P. Heinz, Feb 02 2009

max = 30; A055887 = CoefficientList[1/(2 - 1/QPochhammer[x, x]) + O[x]^(max + 1), x] ; s = 1/Product[(1 - x^n)^A055887[[n + 1]], {n, 1, max}] + O[x]^max; CoefficientList[s, x] // Rest (* Jean-François Alcover, Jan 10 2016 *)

More terms from Alois P. Heinz, Feb 02 2009

cp's OEIS Frontend

A034691 Euler transform of powers of 2 [1,2,4,8,16,...].

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