A025463 Number of partitions of n into 10 positive cubes.
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 0, 1, 0
Offset: 0
Keywords
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Programs
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Maple
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+ `if`(i^3>n, 0, b(n-i^3, i, t-1)))) end: a:= n-> b(n, iroot(n, 3), 10): seq(a(n), n=0..120); # Alois P. Heinz, Dec 21 2018
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Mathematica
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] + If[i^3 > n, 0, b[n - i^3, i, t - 1]]]]; a[n_] := b[n, n^(1/3) // Floor, 10]; a /@ Range[0, 120] (* Jean-François Alcover, Dec 04 2020, after Alois P. Heinz *) Table[Count[IntegerPartitions[n,{10}],?(AllTrue[CubeRoot[#],IntegerQ]&)],{n,0,110}] (* _Harvey P. Dale, Jul 26 2025 *)
Formula
a(n) = [x^n y^10] Product_{k>=1} 1/(1 - y*x^(k^3)). - Ilya Gutkovskiy, Apr 23 2019