A339221 Number of partitions of n into cubes where every part appears at least 2 times.

1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 7, 6, 8, 7, 8, 8, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 10, 9, 11, 10, 11, 11, 11, 11, 12, 11, 13, 13, 13, 14, 14, 14, 15, 14, 16, 15, 16, 16, 16, 16, 17, 16

Offset

0,17

Links

Table of n, a(n) for n=0..95.

Index entries for sequences related to partitions

Formula

G.f.: Product_{k>=1} (1 + x^(2*k^3) / (1 - x^(k^3))).

Example

a(24) = 3 because we have [8, 8, 8], [8, 8, 1, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Mathematica

nmax = 95; CoefficientList[Series[Product[1 + x^(2 k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[n], AllTrue[Surd[#, 3], IntegerQ] && Min[ Tally[ #][[All, 2]]]>1&]], {n, 0, 100}] (* Requires Mathematica version 10 or later *) (* The program takes a long time to run. *) (* Harvey P. Dale, May 03 2021 *)

Crossrefs

Cf. A000578, A003108, A007690, A161090.

Sequence in context: A067754 A194824 A339931 * A025851 A343911 A125688

Adjacent sequences: A339218 A339219 A339220 * A339222 A339223 A339224

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 27 2020

Status

approved