A369574 Expansion of Product_{k>=1} (1 + x^(k^3)) / (1 - x^(k^2)).

1, 2, 2, 2, 3, 4, 4, 4, 6, 9, 10, 10, 12, 15, 16, 16, 19, 24, 27, 28, 31, 36, 39, 40, 44, 52, 58, 62, 68, 76, 82, 86, 93, 104, 114, 122, 134, 148, 158, 166, 179, 196, 210, 223, 242, 265, 282, 295, 315, 342, 365, 384, 412, 447, 476, 498, 527, 566, 602, 632, 670

Offset: 0

Author

Vaclav Kotesovec, Jan 26 2024

Keywords

• nonn

Comments

Convolution of A279329 and A001156.

a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into distinct cubes and P(n-k) is a partition of n-k into squares.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Crossrefs

Cf. A001156, A279329, A280278, A280276.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[(1+x^(k^3))/(1-x^(k^2)), {k, 1, nmax^(1/2)}], {x, 0, nmax}], x]

Formula

a(n) ~ exp((27*zeta(3/2)^(8/9) * (729*(48 + 6*2^(1/3) - 35*2^(2/3)) * Pi^(4/3) * zeta(3/2)^(16/9) * n^(1/3) + 972 * 2^(1/9)*(41 - 59*2^(1/3) + 21*2^(2/3)) * Gamma(4/3) * zeta(4/3) * (Pi*zeta(3/2))^(8/9) * n^(2/9) - 8*2^(2/9)*(-160 + 2^(1/3) + 100*2^(2/3)) * Pi^(4/9) * Gamma(1/3)^2 * zeta(4/3)^2 * n^(1/9)) + 32*(-161 - 99*2^(1/3) + 180*2^(2/3)) * Gamma(1/3)^3 * zeta(4/3)^3) / (26244*(-1 + 2^(1/3))^6 * Pi * zeta(3/2)^2)) * zeta(3/2)^(2/3) * (-198 + 360*2^(1/3) - 161*2^(2/3)) / (8*sqrt(6) * (-1 + 2^(1/3))^9 * Pi^(7/6) * n^(7/6)).