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

0,3

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

Links

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

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A258619 A348757 A380207 * A382078 A381990 A240309

Adjacent sequences: A118804 A118805 A118806 * A118808 A118809 A118810

Keyword

nonn

Author

Emeric Deutsch, Apr 29 2006

Status

approved