A369571 Expansion of Product_{k>=1} (1 + x^(k^3)) * (1 + x^k).

1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 20, 25, 31, 38, 47, 58, 70, 84, 102, 122, 145, 173, 205, 242, 285, 334, 391, 458, 534, 620, 720, 833, 961, 1109, 1276, 1466, 1683, 1926, 2201, 2513, 2863, 3258, 3704, 4203, 4763, 5394, 6098, 6885, 7768, 8752, 9850, 11076, 12439

Offset: 0

Author

Vaclav Kotesovec, Jan 26 2024

Keywords

• nonn

Comments

Convolution of A279329 and A000009.

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 distinct parts.

Links

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

Crossrefs

Cf. A000009, A279329, A280278, A280277.

Programs

Mathematica

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

Formula

a(n) ~ exp(Pi*sqrt(n/3) + (2^(1/3) - 1) * Gamma(1/3) * zeta(4/3) * n^(1/6) / (3^(5/6) * Pi^(1/3))) / (2^(5/2) * 3^(1/4) * n^(3/4)).