A369519 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^(k^3))).

1, 2, 3, 4, 6, 8, 10, 12, 16, 21, 26, 31, 38, 46, 54, 62, 74, 88, 103, 118, 137, 158, 180, 202, 230, 263, 298, 335, 378, 426, 476, 528, 589, 658, 732, 810, 900, 998, 1101, 1208, 1330, 1465, 1608, 1760, 1930, 2116, 2310, 2513, 2738, 2985, 3246, 3521, 3826, 4156

Offset

0,2

Comments

Convolution of A001156 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 into squares and P(n-k) is a partition of n-k into cubes.

Links

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

Formula

a(n) ~ zeta(3/2) * exp(3 * Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3) + 2^(4/9) * Gamma(1/3) * zeta(4/3) * n^(2/9) / (3 * Pi^(1/9) * zeta(3/2)^(2/9)) - 4*2^(2/9) * Gamma(1/3)^2 * zeta(4/3)^2 * n^(1/9) / (243 * Pi^(5/9) * zeta(3/2)^(10/9)) + 16*Gamma(1/3)^3 * zeta(4/3)^3 / (6561 * Pi * zeta(3/2)^2)) / (16 * sqrt(6) * Pi^(5/2) * n^(3/2)) * (1 + (13*2^(7/9) * Gamma(1/3) * zeta(4/3) / (81 * Pi^(4/9) * zeta(3/2)^(8/9)) + 832*2^(7/9) * Gamma(1/3)^4 * zeta(4/3)^4 / (1594323 * Pi^(13/9) * zeta(3/2)^(26/9))) / n^(1/9) + (692224 * 2^(5/9) * Gamma(1/3)^8 * zeta(4/3)^8 / (2541865828329 * Pi^(26/9) * zeta(3/2)^(52/9)) - 128 * 2^(5/9) * Gamma(1/3)^5 * zeta(4/3)^5 / (4782969 * Pi^(17/9) * zeta(3/2)^(34/9)) + 65*2^(5/9) * Gamma(1/3)^2 * zeta(4/3)^2 / (2187*Pi^(8/9) * zeta(3/2)^(16/9)))/n^(2/9)).

Mathematica

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

Crossrefs

Cf. A001156, A003108.

Cf. A087153, A131799, A264393, A369520, A369579.

Sequence in context: A088881 A020697 A175381 * A213623 A216365 A034287

Adjacent sequences: A369516 A369517 A369518 * A369520 A369521 A369522

Keyword

nonn

Author

Vaclav Kotesovec, Jan 25 2024

Status

approved