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A103446
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Unlabeled analog of A025168.
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4
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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
Starting (1, 3, 8, 21, 54, 137,...), = row sums of triangle A137151 - Gary W. Adamson, Jan 23 2008
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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
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FORMULA
| O.g.f.: exp( Sum_{n>=1} sigma(n)*x^n/(1-x)^n/n ) - 1. [From 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. [From 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)) [From Kruchinin Vladimir, Aug 10 2010]
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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]]}.
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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.
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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))} [From 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))} [From Paul D. Hanna, Feb 04 2012]
(PARI) Vec(1/eta('x/(1-'x)+O('x^66))) [Joerg Arndt, Jul 30 2011]
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CROSSREFS
| Cf. A025168, A034691, A050351, A137151, A185003 (log).
Sequence in context: A027930 A038200 A030015 * A094723 A127358 A077849
Adjacent sequences: A103443 A103444 A103445 * A103447 A103448 A103449
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KEYWORD
| nonn,changed
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AUTHOR
| Thomas Wieder (wieder.thomas(AT)t-online.de), Feb 06 2005; revised Feb 20 2006
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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 (njas(AT)research.att.com) - May 18 2007
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