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A218495 Number of partitions of n^2 into positive cubes.

%I A218495 #33 Jun 05 2017 17:34:11
%S A218495 1,1,1,2,3,4,7,10,17,26,39,58,89,133,195,289,420,610,875,1253,1778,
%T A218495 2514,3527,4937,6879,9516,13115,18012,24625,33503,45432,61402,82677,
%U A218495 110913,148286,197722,262768,348100,459791,605780,795874,1042791,1362800,1776777
%N A218495 Number of partitions of n^2 into positive cubes.
%C A218495 a(n) = A003108(A000290(n)).
%H A218495 Alois P. Heinz, <a href="/A218495/b218495.txt">Table of n, a(n) for n = 0..1000</a>
%F A218495 a(n) ~ exp(4 * (Gamma(1/3)*Zeta(4/3))^(3/4) * sqrt(n) / 3^(3/2)) * (Gamma(1/3)*Zeta(4/3))^(3/4) / (24*Pi^2*n^(5/2)) [after Hardy & Ramanujan, 1917]. - _Vaclav Kotesovec_, Apr 10 2017
%F A218495 a(n) = [x^(n^2)] Product_{k>=1} 1/(1 - x^(k^3)). - _Ilya Gutkovskiy_, Jun 05 2017
%e A218495 n=5: number of partitions of 25 into parts of {1, 8}:
%e A218495 a(5) = #{8+8+8+1, 8+8+9x1, 8+17x1, 25x1} = 4;
%e A218495 n=6: number of partitions of 36 into parts of {1, 8, 27}:
%e A218495 a(6) = #{27+8+1, 27+9x1, 4x8+4x1, 3x8+12x1, 8+8+20x1, 8+28x1, 36x1} = 7;
%e A218495 n=7: number of partitions of 49 into parts of {1, 8, 27}:
%e A218495 a(7) = #{27+8+8+6x1, 27+8+14x1, 27+22x1, 6x8+1, 5x8+9x1, 4x8+17x1, 3x8+25x1, 8+8+33x1, 8+41x1, 49x1} = 10.
%p A218495 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A218495       b(n, i-1) +`if`(i^3>n, 0, b(n-i^3, i)))
%p A218495     end:
%p A218495 a:= n-> b(n^2, iroot(n^2, 3)):
%p A218495 seq(a(n), n=0..60);  # _Alois P. Heinz_, Nov 08 2012
%t A218495 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i-1] + If[i^3>n, 0, b[n - i^3, i]]]; a[n_] := b[n^2, n^(2/3) // Floor]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Nov 11 2015, after _Alois P. Heinz_ *)
%o A218495 (Haskell) a218495 = p (tail a000578_list) . (^ 2) where
%o A218495    p _          0 = 1
%o A218495    p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o A218495 (PARI) a(n) = {my(nb=0); forpart(p=n^2, nb += (sum(k=1, #p, ispower(p[k], 3)) == #p);); nb;} \\ _Michel Marcus_, Jun 02 2015
%o A218495 (PARI) ok(p)=for(i=1,#p,if(!ispower(p[i],3),return(0)));1
%o A218495 a(n)=my(s=1);for(i=8,n^2,forpart(p=i,s+=ok(p),[8,sqrtnint(i,3)^3]));s \\ _Charles R Greathouse IV_, Jun 02 2015
%Y A218495 Cf. A000578, A218494.
%K A218495 nonn
%O A218495 0,4
%A A218495 _Reinhard Zumkeller_, Oct 31 2012
%E A218495 More terms from _Alois P. Heinz_, Nov 08 2012