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A027336
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Number of partitions of n that do not contain 2 as a part.
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16
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1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 35, 45, 58, 75, 96, 121, 154, 193, 242, 302, 375, 463, 573, 703, 861, 1052, 1282, 1555, 1886, 2277, 2745, 3301, 3961, 4740, 5667, 6754, 8038, 9548, 11323, 13398, 15836, 18678, 22001, 25873, 30383, 35620, 41715, 48771
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Also number of partitions of n into parts with at most one 1. - Reinhard Zumkeller, Oct 25 2004
Also number of partitions of n into parts with at least half of the parts having size 1; equivalently (by duality) number of partitions of n where the large part is at least twice as big as the second largest part. - Frank Adams-Watters, Jun 08 2005
Also number of 2-regular not necessarily connected graphs with loops allowed but no multiple edges.
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LINKS
| P. Chinn and S. Heubach, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 6 (2003), no. 2, Article 03.2.3.
Jerome Kelleher and Barry O'Sullivan, Generating All Partitions: A Comparison Of Two Encodings, p.24, (2009), arXiv:0909.2331v1 [From Peter Luschny, Oct 24 2010]
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FORMULA
| G.f.: (1-x^2)*prod(m>=1, 1/(1-x^m) ).
a(n)=p(n)-p(n-2) for n>=2, where p(n) are the partition numbers (A000041); follows at once from the g.f. - Emeric Deutsch, Feb 18 2006
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MAPLE
| with(combinat): a:=proc(n) if n=0 then 1 elif n=1 then 1 else numbpart(n)-numbpart(n-2) fi end: seq(a(n), n=0..49); - Emeric Deutsch, Feb 18 2006
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MATHEMATICA
| a[n_] = PartitionsP[n] - PartitionsP[n-2]; a /@ Range[0, 49] (* From Jean-François Alcover, Jul 13 2011, after E. Deutsch *)
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PROG
| (PARI) a(n)=if(n<0, 0, polcoeff((1-x^2)/eta(x+x*O(x^n)), n))
(MAGMA) A41 := func<n|n ge 0 select NumberOfPartitions(n) else 0>;
[A41(n)-A41(n-2):n in [0..49]];
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CROSSREFS
| Cf. A027337.
Pairwise sums of sequence A002865 (partitions in which the least part is at least 2).
a(n)=A000041(n)-A000041(n-2).
2-regular not necessarily connected graphs: A008483 (simple graphs), A000041 (multigraphs with loops allowed), A002865 (multigraphs with loops forbidden), A027336 (graphs with loops allowed but no multiple edges).
Sequence in context: A035980 A035990 A036001 * A023434 A087192 A188917
Adjacent sequences: A027333 A027334 A027335 * A027337 A027338 A027339
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KEYWORD
| nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 10 2002
Regular graphs comment and cross references, and Magma code from Jason Kimberley (Jason.Kimberley(AT)newcastle.edu.au), Jan 05 2011
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