A103446 Unlabeled analog of A025168.

0, 1, 3, 8, 21, 54, 137, 344, 856, 2113, 5179, 12614, 30548, 73595, 176455, 421215, 1001388, 2371678, 5597245, 13166069, 30873728, 72185937, 168313391, 391428622, 908058205, 2101629502, 4853215947, 11183551059, 25718677187, 59030344851, 135237134812

Offset

0,3

Comments

Or, if the initial 0 is omitted, this is the binomial transform of the partition numbers p(1), p(2), ... = 1, 2, 3, 5, 7, 11, 15, 22, 30, ... (A000041 without the initial 1).

The most precise definition of this sequence is the Maple combstruct command given below. See the first Wieder link for further details.

Sequence appears to have a rational o.g.f. - Ralf Stephan, May 18 2007

For n>0, row sums of triangle A137151. - Gary W. Adamson, Jan 23 2008

a(n) = A218482(n) for n>=1; see A218482 for more formulas.

Links

Table of n, a(n) for n=0..30.

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

Thomas Wieder, Expanded definitions of A103446 and A025168

Formula

O.g.f.: exp( Sum_{n>=1} sigma(n)*x^n/(1-x)^n/n ) - 1. - Paul D. Hanna, Apr 21 2010

O.g.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k)*sigma(k) ) - 1. - Paul D. Hanna, Feb 04 2012

O.g.f. P(x/(1-x)), where P(x) is the o.g.f. for number of partitions (A000041) a(n)=sum_{k=1,n} ( binomial(n-1,k-1)*A000041(k)). - Vladimir Kruchinin, Aug 10 2010

a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-2) / (n*sqrt(3)). - Vaclav Kotesovec, Jun 25 2015

Example

Let {} denote a set, [] a list and Z an unlabeled element.
a(3) = 8 because we have {[[Z]],[[Z]],[[Z]]}, {[[Z],[Z]],[[Z]]}, {[[Z],[Z],[Z]]}, {[[Z],[Z,Z]]}, {[[Z,Z],[Z]]}, {[[Z,Z]],[[Z]]}, {[[Z]],[[Z,Z]]}, {[[Z,Z,Z]]}.

Maple

with(combstruct); SubSetSeqU := [T, {T=Subst(U, S), S=Set(U, card>=1), U=Sequence(Z, card>=1)}, unlabeled]; [seq(count(SubSetSeqU, size=n), n=0..30)];
allstructs(SubSetSeq, size=3); # to get the structures for n=3 - this output is shown in the example lines.

Mathematica

Flatten[{0, Table[Sum[Binomial[n-1, k]*PartitionsP[k+1], {k, 0, n-1}], {n, 1, 30}]}] (* Vaclav Kotesovec, Jun 25 2015 *)

Prog

(PARI) {a(n)=if(n<1, 0, polcoeff(exp(sum(m=1, n, sigma(m)*x^m/(1-x+x*O(x^n))^m/m)), n))} \\ Paul D. Hanna, Apr 21 2010
(PARI) {a(n)=if(n<1, 0, polcoeff(exp(sum(m=1, n, x^m/m*sum(k=1, m, binomial(m, k)*sigma(k)))+x*O(x^n)), n))} \\ Paul D. Hanna, Feb 04 2012
(PARI) Vec(1/eta('x/(1-'x)+O('x^66))) \\ Joerg Arndt, Jul 30 2011

Crossrefs

Cf. A025168, A034691, A050351, A137151, A185003 (log), A218482.

Sequence in context: A291039 A030015 A318567 * A218482 A094723 A127358

Adjacent sequences: A103443 A103444 A103445 * A103447 A103448 A103449

Keyword

nonn

Author

Thomas Wieder, Feb 06 2005; revised Feb 20 2006

Extensions

I can confirm that the terms shown are the binomial transform of the partition sequence 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, ... (A000041 without the a(0) term). - N. J. A. Sloane, May 18 2007

Status

approved