A342229 Total sum of parts which are cubes in all partitions of n.

0, 1, 2, 4, 7, 12, 19, 30, 53, 75, 113, 163, 235, 328, 461, 628, 868, 1163, 1564, 2069, 2743, 3578, 4674, 6036, 7795, 9962, 12728, 16151, 20441, 25714, 32290, 40332, 50292, 62405, 77288, 95339, 117382, 143987, 176298, 215168, 262121, 318385, 386043, 466838, 563577, 678712

Offset

0,3

Links

Table of n, a(n) for n=0..45.

Formula

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

a(n) = Sum_{k=1..n} A113061(k) * A000041(n-k).

Example

For n = 4 we have:
--------------------------------
Partitions        Sum of parts
.               which are cubes
--------------------------------
4 ................... 0
3 + 1 ............... 1
2 + 2 ............... 0
2 + 1 + 1 ........... 2
1 + 1 + 1 + 1 ....... 4
--------------------------------
Total ............... 7
So a(4) = 7.

Mathematica

nmax = 45; CoefficientList[Series[Sum[k^3 x^(k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}]/Product[(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[DivisorSum[k, # &, IntegerQ[#^(1/3)] &] PartitionsP[n - k], {k, 1, n}], {n, 0, 45}]

Crossrefs

Cf. A000041, A000578, A066186, A113061, A264392, A342228.

Sequence in context: A102346 A333148 A343661 * A326080 A287525 A244472

Adjacent sequences: A342226 A342227 A342228 * A342230 A342231 A342232

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 06 2021

Status

approved