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A027336 Number of partitions of n that do not contain 2 as a part. 39
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; text; internal format)
OFFSET
0,4
COMMENTS
Pairwise sums of sequence A002865 (partitions in which the least part is at least 2).
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. - Franklin T. Adams-Watters, Jun 08 2005
Also number of 2-regular not necessarily connected graphs with loops allowed but no multiple edges. - Jason Kimberley, Jan 05 2011
LINKS
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
V. Jelinek, T. Mansour, and M. Shattuck, On multiple pattern avoiding set partitions, Advances in Applied Mathematics Volume 50, Issue 2, February 2013, Pages 292-326. - N. J. A. Sloane, Jan 01 2013
Jerome Kelleher and Barry O'Sullivan, Generating All Partitions: A Comparison Of Two Encodings, arXiv:0909.2331 [cs.DS], 2009-2014. [Peter Luschny, Oct 24 2010]
Krishna Menon and Anurag Singh, Pattern avoidance and dominating compositions, arXiv:2104.07274 [math.CO], 2021.
Mircea Merca, Fast algorithm for generating ascending compositions, arXiv:1903.10797 [math.CO], 2019.
FORMULA
G.f.: (1 - x^2)*Product_{m>=1} 1/(1 - x^m).
a(n) = A000041(n) - A000041(n-2).
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
a(n) ~ exp(sqrt(2*n/3)*Pi)*Pi / (6*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 25*Pi/(24*sqrt(6)))/sqrt(n) + (25/8 + 9/(2*Pi^2) + 817*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016
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
MATHEMATICA
a[n_] = PartitionsP[n] - PartitionsP[n-2]; a /@ Range[0, 49] (* Jean-François Alcover, Jul 13 2011, after Emeric Deutsch *)
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]]; // Jason Kimberley, Jan 05 2011
CROSSREFS
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). - Jason Kimberley, Jan 05 2011
Column k=1 of A292622.
Sequence in context: A035980 A035990 A036001 * A237830 A023434 A353035
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Benoit Cloitre, Dec 10 2002
STATUS
approved

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Last modified April 25 04:42 EDT 2024. Contains 371964 sequences. (Running on oeis4.)