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

%I A118807 #16 Jun 12 2025 08:00:09
%S A118807 1,1,2,2,5,6,9,12,19,24,34,43,62,77,105,132,177,220,287,356,462,570,
%T A118807 723,888,1121,1370,1705,2074,2570,3111,3816,4601,5617,6743,8170,9777,
%U A118807 11794,14058,16858,20029,23932,28334,33692,39772,47133,55468,65471,76840
%N A118807 Number of partitions of n having no parts with multiplicity 3.
%C A118807 Column 0 of A118806.
%C A118807 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
%H A118807 Alois P. Heinz, <a href="/A118807/b118807.txt">Table of n, a(n) for n = 0..1000</a>
%F A118807 G.f.: Product_{j>=1} (1 + x^j + x^(2j) + x^(4j)/(1-x^j)).
%F A118807 a(n) = A000041(n) - A183560(n) = A183568(n,0) - A183568(n,3). - _Alois P. Heinz_, Oct 09 2011
%F A118807 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
%e A118807 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.
%p A118807 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);
%t A118807 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 *)
%Y A118807 Cf. A000041, A118806, A007690, A116645, A183560, A183568.
%K A118807 nonn
%O A118807 0,3
%A A118807 _Emeric Deutsch_, Apr 29 2006