A027336 Number of partitions of n that do not contain 2 as a part.

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

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

Table of n, a(n) for n=0..49.

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

Cf. A000041, A002865, A027337.

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

Adjacent sequences: A027333 A027334 A027335 * A027337 A027338 A027339

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Benoit Cloitre, Dec 10 2002

Status

approved