A116645 Number of partitions of n having no doubletons. By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] of 18 has two doubletons, shown between parentheses).

1, 1, 1, 3, 3, 5, 8, 10, 13, 20, 26, 33, 46, 58, 75, 101, 125, 157, 206, 253, 317, 403, 494, 608, 760, 926, 1131, 1393, 1685, 2038, 2487, 2985, 3585, 4331, 5168, 6172, 7392, 8771, 10410, 12382, 14622, 17258, 20400, 23975, 28159, 33115, 38739, 45298, 53000

Offset: 0

Author

Emeric Deutsch and Vladeta Jovovic, Feb 20 2006

Keywords

• nonn

Comments

Number of partitions of n having no part that appears exactly twice.

Infinite convolution product of [1,1,0,1,1,1,1,1,1,1] aerated n-1 times. I.e., [1,1,0,1,1,1,1,1,1,1] * [1,0,1,0,0,0,1,0,1,0] * [1,0,0,1,0,0,0,0,0,1] * ... - Mats Granvik, Gary W. Adamson, Aug 07 2009

Examples

			a(4) = 3 because we have [4],[3,1] and [1,1,1,1] (the partitions [2,2] and [2,1,1] do not qualify since each of them has a doubleton).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Column 0 of A116644.

Cf. A000041, A007690, A116595, A183559, A183568.

Programs

Maple

h:=product((1-x^(2*j)+x^(3*j))/(1-x^j),j=1..60): hser:=series(h,x=0,60): seq(coeff(hser,x,n),n=0..56);

Mathematica

nn=48;CoefficientList[Series[Product[1/(1-x^i)-x^(2i),{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 30 2013 *)

Formula

G.f.: Product_{j>=1} (1-x^(2j)+x^(3j))/(1-x^j).

G.f. for the number of partitions of n having no part that appears exactly m times is Product_{k>0} (1/(1-x^k)-x^(m*k)).

a(n) = A000041(n) - A183559(n) = A183568(n,0) - A183568(n,2). - Alois P. Heinz, Oct 09 2011

a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-2*x) + exp(-4*x)) dx = 0.64673501839556449802623523266221107725058748270577037891948... - Vaclav Kotesovec, Jun 12 2025