A298434 Expansion of Product_{k>=1} 1/(1 - x^(k^3))^2.

1, 2, 3, 4, 5, 6, 7, 8, 11, 14, 17, 20, 23, 26, 29, 32, 38, 44, 50, 56, 62, 68, 74, 80, 90, 100, 110, 122, 134, 146, 158, 170, 187, 204, 221, 242, 263, 284, 305, 326, 353, 380, 407, 440, 473, 506, 539, 572, 612, 652, 692, 740, 788, 836, 887, 938, 997, 1056, 1115, 1184, 1253

Offset

0,2

Comments

Number of partitions of n into cubes of 2 kinds.

Self-convolution of A003108.

Links

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

Index entries for related partition-counting sequences

Formula

G.f.: Product_{k>=1} 1/(1 - x^(k^3))^2.

a(n) ~ exp(2^(11/4) * (Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3) * Zeta(4/3))^(9/8) / (2^(27/8) * 3^(7/4) * Pi^(7/2) * n^(13/8)). - Vaclav Kotesovec, Apr 08 2018

Example

a(8) = 11 because we have [8a], [8b], [1a, 1a, 1a, 1a, 1a, 1a, 1a, 1a], [1a, 1a, 1a, 1a, 1a, 1a, 1a, 1b], [1a, 1a, 1a, 1a, 1a, 1a, 1b, 1b], [1a, 1a, 1a, 1a, 1a, 1b, 1b, 1b], [1a, 1a, 1a, 1a, 1b, 1b, 1b, 1b], [1a, 1a, 1a, 1b, 1b, 1b, 1b, 1b], [1a, 1a, 1b, 1b, 1b, 1b, 1b, 1b], [1a, 1b, 1b, 1b, 1b, 1b, 1b, 1b] and [1b, 1b, 1b, 1b, 1b, 1b, 1b, 1b].

Mathematica

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(k^3))^2, {k, 1, Floor[nmax^(1/3) + 1]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 08 2018 *)

Crossrefs

Cf. A000578, A000712, A003108, A279225.

Sequence in context: A351914 A258946 A008815 * A200371 A246025 A211699

Adjacent sequences: A298431 A298432 A298433 * A298435 A298436 A298437

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 19 2018

Status

approved