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
Keywords
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
Programs
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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
Comments