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A025463 Number of partitions of n into 10 positive cubes.

%I A025463 #20 Jul 26 2025 18:53:34
%S A025463 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,
%T A025463 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,
%U A025463 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
%N A025463 Number of partitions of n into 10 positive cubes.
%H A025463 Alois P. Heinz, <a href="/A025463/b025463.txt">Table of n, a(n) for n = 0..65536</a>
%H A025463 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A025463 a(n) = [x^n y^10] Product_{k>=1} 1/(1 - y*x^(k^3)). - _Ilya Gutkovskiy_, Apr 23 2019
%p A025463 b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
%p A025463       `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A025463       `if`(i^3>n, 0, b(n-i^3, i, t-1))))
%p A025463     end:
%p A025463 a:= n-> b(n, iroot(n, 3), 10):
%p A025463 seq(a(n), n=0..120);  # _Alois P. Heinz_, Dec 21 2018
%t A025463 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]]]];
%t A025463 a[n_] := b[n, n^(1/3) // Floor, 10];
%t A025463 a /@ Range[0, 120] (* _Jean-François Alcover_, Dec 04 2020, after _Alois P. Heinz_ *)
%t A025463 Table[Count[IntegerPartitions[n,{10}],_?(AllTrue[CubeRoot[#],IntegerQ]&)],{n,0,110}] (* _Harvey P. Dale_, Jul 26 2025 *)
%Y A025463 Cf. A000578 (cubes).
%Y A025463 Column k=10 of A320841.
%K A025463 nonn
%O A025463 0,74
%A A025463 _David W. Wilson_