A303904 Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^(k^3)).

1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13

Offset: 0

Author

Ilya Gutkovskiy, May 02 2018

Keywords

• nonn

Comments

Partial sums of A279329.

Links

• Index entries for sequences related to partitions
• Index entries for sequences related to sums of cubes

Crossrefs

Cf. A000578, A003997, A036469, A038348, A248801, A279329, A302834.

Programs

Maple

b:= proc(n, i) option remember; `if`(n<0, 0,
     `if`(n=0, 1, `if`(n>i^2*(i+1)^2/4, 0, (t->
       b(t, min(t, i-1)))(n-i^3)+b(n, i-1))))
    end:
a:= proc(n) option remember; `if`(n<0, 0,
       b(n, iroot(n, 3))+a(n-1))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, May 02 2018

Mathematica

nmax = 91; CoefficientList[Series[1/(1 - x) Product[1 + x^k^3, {k, 1, Floor[nmax^(1/3) + 1]}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(2^(7/4) * ((2^(1/3) - 1) * Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * 3^(5/4) / (2^(15/8) * sqrt(Pi) * ((2^(1/3) - 1) * Gamma(1/3) * Zeta(4/3))^(3/8) * n^(1/8)). - Vaclav Kotesovec, May 04 2018