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

Author

Thomas Wieder, Feb 06 2005; revised Feb 20 2006

Keywords

• nonn

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.

Examples

			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]]}.
		

Links

• 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

Crossrefs

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

Programs

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 *)

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

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

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