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A369576
Expansion of Product_{k>=1} 1 / ((1 - x^k) * (1 - x^(k^2)) * (1 - x^(k^3))).
2
1, 3, 7, 14, 27, 48, 82, 134, 215, 336, 515, 773, 1145, 1670, 2406, 3423, 4824, 6730, 9310, 12768, 17385, 23499, 31559, 42111, 55876, 73726, 96784, 126418, 164375, 212772, 274277, 352125, 450365, 573891, 728765, 922305, 1163530, 1463287, 1834842, 2294128, 2860538
OFFSET
0,2
COMMENTS
Convolution of A000041 and A001156 and A003108.
Convolution of A369520 and A003108.
a(n) is the number of triples (R(r), S(s), T(t)) where r + s + t = n, and R(k) is a partition of k, S(k) a partition of k into squares, and T(k) a partition of k into cubes.
LINKS
FORMULA
a(n) ~ exp(Pi*sqrt(2*n/3) + 3^(1/4) * zeta(3/2) * n^(1/4) / 2^(3/4) + 6^(1/6) * Gamma(4/3) * zeta(4/3) * n^(1/6) / Pi^(1/3) - 3*zeta(3/2)^2 / (32*Pi)) / (96 * Pi^(3/2) * n^(3/2)) * (1 - Gamma(1/3) * zeta(4/3) * zeta(3/2) / (12 * 6^(1/12) * Pi^(4/3) * n^(1/12))).
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[1/((1-x^k)*(1-x^(k^2))*(1-x^(k^3))), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Jan 26 2024
STATUS
approved