A369575 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(k^2)) * (1 + x^(k^3)).

1, 3, 4, 5, 8, 12, 16, 21, 28, 38, 51, 65, 82, 105, 133, 166, 206, 254, 312, 382, 464, 561, 677, 813, 972, 1160, 1380, 1636, 1935, 2281, 2682, 3148, 3683, 4297, 5008, 5826, 6761, 7832, 9055, 10451, 12045, 13855, 15909, 18246, 20895, 23891, 27282, 31110, 35427

Offset: 0

Author

Vaclav Kotesovec, Jan 26 2024

Keywords

• nonn

Comments

Convolution of A000009 and A033461 and A279329.

Convolution of A369570 and A279329.

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 into distinct parts, S(k) a partition of k into distinct squares, and T(k) a partition of k into distinct cubes.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000009, A033461, A279329, A369570, A369571, A369572.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[(1+x^k)*(1+x^(k^2))*(1+x^(k^3)), {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)) + 3^(1/4)*(sqrt(2) - 1) * zeta(3/2) * n^(1/4)/2 + 3*(2*sqrt(2) - 3) * zeta(3/2)^2 / (32*Pi)) / (8*3^(1/4)*n^(3/4)).