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

Author

Ilya Gutkovskiy, Mar 06 2021

Keywords

• nonn

Examples

			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.
		

Crossrefs

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

Programs

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}]

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).