A369579 Expansion of Product_{k>=1} 1 / ((1 - x^k) * (1 - x^(k^3))).

1, 2, 4, 7, 12, 19, 30, 45, 68, 99, 143, 202, 284, 392, 538, 729, 983, 1311, 1740, 2289, 2998, 3898, 5046, 6492, 8321, 10607, 13472, 17032, 21460, 26927, 33682, 41975, 52160, 64600, 79790, 98255, 120690, 147836, 180662, 220217, 267841, 324999, 393539, 475496, 573403

Offset: 0

Author

Vaclav Kotesovec, Jan 26 2024

Keywords

• nonn

Comments

Convolution of A000041 and A003108.

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

Links

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

Crossrefs

Cf. A000041, A001156, A003108, A280278, A369519, A369520.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[1/((1-x^k)*(1-x^(k^3))), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(2*n/3) + 6^(1/6) * Gamma(4/3) * zeta(4/3) * n^(1/6) / Pi^(1/3)) / (2^(15/4) * 3^(3/4) * Pi * n^(5/4)) * (1 - Gamma(1/3)^2 * zeta(4/3)^2 / (54 * 6^(1/6) * Pi^(5/3) * n^(1/6))).