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%I #43 Jun 03 2017 04:52:50
%S 0,1,3,8,21,54,137,344,856,2113,5179,12614,30548,73595,176455,421215,
%T 1001388,2371678,5597245,13166069,30873728,72185937,168313391,
%U 391428622,908058205,2101629502,4853215947,11183551059,25718677187,59030344851,135237134812
%N Unlabeled analog of A025168.
%C 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).
%C The most precise definition of this sequence is the Maple combstruct command given below. See the first Wieder link for further details.
%C Sequence appears to have a rational o.g.f. - _Ralf Stephan_, May 18 2007
%C For n>0, row sums of triangle A137151. - _Gary W. Adamson_, Jan 23 2008
%C a(n) = A218482(n) for n>=1; see A218482 for more formulas.
%H N. J. A. Sloane and Thomas Wieder, <a href="http://arXiv.org/abs/math.CO/0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 83-89.
%H Thomas Wieder, <a href="/A103446/a103446_1.txt">Expanded definitions of A103446 and A025168</a>
%F O.g.f.: exp( Sum_{n>=1} sigma(n)*x^n/(1-x)^n/n ) - 1. - _Paul D. Hanna_, Apr 21 2010
%F 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
%F 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
%F a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-2) / (n*sqrt(3)). - _Vaclav Kotesovec_, Jun 25 2015
%e Let {} denote a set, [] a list and Z an unlabeled element.
%e 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]]}.
%p 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)];
%p allstructs(SubSetSeq,size=3); # to get the structures for n=3 - this output is shown in the example lines.
%t Flatten[{0, Table[Sum[Binomial[n-1,k]*PartitionsP[k+1],{k,0,n-1}],{n,1,30}]}] (* _Vaclav Kotesovec_, Jun 25 2015 *)
%o (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
%o (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
%o (PARI) Vec(1/eta('x/(1-'x)+O('x^66))) \\ _Joerg Arndt_, Jul 30 2011
%Y Cf. A025168, A034691, A050351, A137151, A185003 (log), A218482.
%K nonn
%O 0,3
%A _Thomas Wieder_, Feb 06 2005; revised Feb 20 2006
%E 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
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