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

Author

Clark Kimberling

Keywords

• nonn

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.

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.

Programs

Magma

A41 := func;
[A41(n)-A41(n-2):n in [0..49]]; // Jason Kimberley, Jan 05 2011

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 *)

PARI

a(n)=if(n<0,0,polcoeff((1-x^2)/eta(x+x*O(x^n)),n))

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

Extensions

More terms from Benoit Cloitre, Dec 10 2002