A302834 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^3)).

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 52, 56, 60, 65, 70, 75, 80, 85, 91, 97, 103, 110, 117, 124, 131, 138, 146, 154, 162, 171, 180, 189, 198, 207, 217, 227, 237, 248, 259, 270, 282, 294, 307, 320, 333, 347, 361, 375, 390, 405, 422

Offset: 0

Author

Ilya Gutkovskiy, Apr 13 2018

Keywords

• nonn

Comments

Partial sums of A003108.

Number of partitions of n into cubes if there are two kinds of 1's.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000
• Index entries for sequences related to partitions
• Index entries for sequences related to sums of cubes

Crossrefs

Cf. A000070, A000578, A003108, A078128, A298434, A302833.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
      b(n, i-1)+ `if`(i^3>n, 0, b(n-i^3, i)))
    end:
a:= n-> b(n, iroot(n, 3)):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 13 2018

Mathematica

nmax = 64; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k^3), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 64; CoefficientList[Series[1/(1 - x) Sum[x^j^3/Product[(1 - x^k^3), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: (1/(1 - x))*Sum_{j>=0} x^(j^3)/Product_{k=1..j} (1 - x^(k^3)).

From Vaclav Kotesovec, Apr 13 2018: (Start)

a(n) ~ sqrt(3) * exp(4*(Gamma(1/3)*Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) / (8 * Pi^2 * sqrt(n)).

a(n) ~ 3^(3/2) * n^(3/4) / (Gamma(1/3)*Zeta(4/3))^(3/4) * A003108(n). (End)