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%I #4 Mar 07 2021 03:56:01
%S 0,1,2,4,7,12,19,30,53,75,113,163,235,328,461,628,868,1163,1564,2069,
%T 2743,3578,4674,6036,7795,9962,12728,16151,20441,25714,32290,40332,
%U 50292,62405,77288,95339,117382,143987,176298,215168,262121,318385,386043,466838,563577,678712
%N Total sum of parts which are cubes in all partitions of n.
%F G.f.: Sum_{k>=1} k^3*x^(k^3)/(1 - x^(k^3)) / Product_{j>=1} (1 - x^j).
%F a(n) = Sum_{k=1..n} A113061(k) * A000041(n-k).
%e For n = 4 we have:
%e --------------------------------
%e Partitions Sum of parts
%e . which are cubes
%e --------------------------------
%e 4 ................... 0
%e 3 + 1 ............... 1
%e 2 + 2 ............... 0
%e 2 + 1 + 1 ........... 2
%e 1 + 1 + 1 + 1 ....... 4
%e --------------------------------
%e Total ............... 7
%e So a(4) = 7.
%t 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]
%t Table[Sum[DivisorSum[k, # &, IntegerQ[#^(1/3)] &] PartitionsP[n - k], {k, 1, n}], {n, 0, 45}]
%Y Cf. A000041, A000578, A066186, A113061, A264392, A342228.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Mar 06 2021
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