A118807 Number of partitions of n having no parts with multiplicity 3.

1, 1, 2, 2, 5, 6, 9, 12, 19, 24, 34, 43, 62, 77, 105, 132, 177, 220, 287, 356, 462, 570, 723, 888, 1121, 1370, 1705, 2074, 2570, 3111, 3816, 4601, 5617, 6743, 8170, 9777, 11794, 14058, 16858, 20029, 23932, 28334, 33692, 39772, 47133, 55468, 65471, 76840

Offset: 0

Author

Emeric Deutsch, Apr 29 2006

Keywords

• nonn

Comments

Column 0 of A118806.

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

Examples

			a(6) = 9 because among the 11 (=A000041(6)) partitions of 6 only [2,2,2] and [3,1,1,1] have parts with multiplicity 3.
		

Links

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

Crossrefs

Cf. A000041, A118806, A007690, A116645, A183560, A183568.

Programs

Maple

g:=product(1+x^j+x^(2*j)+x^(4*j)/(1-x^j),j=1..60): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..50);

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k) + x^(4*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)

Formula

G.f.: Product_{j>=1} (1 + x^j + x^(2j) + x^(4j)/(1-x^j)).

a(n) = A000041(n) - A183560(n) = A183568(n,0) - A183568(n,3). - 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(-3*x) + exp(-5*x)) dx = 0.73597677748514060768682570953508781551028221145343244320009... - Vaclav Kotesovec, Jun 12 2025