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A342229
Total sum of parts which are cubes in all partitions of n.
1
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
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 06 2021
STATUS
approved