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A369519 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^(k^3))).

%I A369519 #15 Jan 27 2024 10:42:13
%S A369519 1,2,3,4,6,8,10,12,16,21,26,31,38,46,54,62,74,88,103,118,137,158,180,
%T A369519 202,230,263,298,335,378,426,476,528,589,658,732,810,900,998,1101,
%U A369519 1208,1330,1465,1608,1760,1930,2116,2310,2513,2738,2985,3246,3521,3826,4156
%N A369519 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^(k^3))).
%C A369519 Convolution of A001156 and A003108.
%C A369519 a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into squares and P(n-k) is a partition of n-k into cubes.
%H A369519 Vaclav Kotesovec, <a href="/A369519/b369519.txt">Table of n, a(n) for n = 0..10000</a>
%F A369519 a(n) ~ zeta(3/2) * exp(3 * Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3) + 2^(4/9) * Gamma(1/3) * zeta(4/3) * n^(2/9) / (3 * Pi^(1/9) * zeta(3/2)^(2/9)) - 4*2^(2/9) * Gamma(1/3)^2 * zeta(4/3)^2 * n^(1/9) / (243 * Pi^(5/9) * zeta(3/2)^(10/9)) + 16*Gamma(1/3)^3 * zeta(4/3)^3 / (6561 * Pi * zeta(3/2)^2)) / (16 * sqrt(6) * Pi^(5/2) * n^(3/2)) * (1 + (13*2^(7/9) * Gamma(1/3) * zeta(4/3) / (81 * Pi^(4/9) * zeta(3/2)^(8/9)) + 832*2^(7/9) * Gamma(1/3)^4 * zeta(4/3)^4 / (1594323 * Pi^(13/9) * zeta(3/2)^(26/9))) / n^(1/9) + (692224 * 2^(5/9) * Gamma(1/3)^8 * zeta(4/3)^8 / (2541865828329 * Pi^(26/9) * zeta(3/2)^(52/9)) - 128 * 2^(5/9) * Gamma(1/3)^5 * zeta(4/3)^5 / (4782969 * Pi^(17/9) * zeta(3/2)^(34/9)) + 65*2^(5/9) * Gamma(1/3)^2 * zeta(4/3)^2 / (2187*Pi^(8/9) * zeta(3/2)^(16/9)))/n^(2/9)).
%t A369519 nmax=100; CoefficientList[Series[Product[1/(1-x^(k^2))/(1-x^(k^3)), {k, 1, nmax^(1/2)}], {x, 0, nmax}], x]
%Y A369519 Cf. A001156, A003108.
%Y A369519 Cf. A087153, A131799, A264393, A369520, A369579.
%K A369519 nonn
%O A369519 0,2
%A A369519 _Vaclav Kotesovec_, Jan 25 2024