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

1, 1, 1, 2, 3, 4, 7, 10, 17, 26, 39, 58, 89, 133, 195, 289, 420, 610, 875, 1253, 1778, 2514, 3527, 4937, 6879, 9516, 13115, 18012, 24625, 33503, 45432, 61402, 82677, 110913, 148286, 197722, 262768, 348100, 459791, 605780, 795874, 1042791, 1362800, 1776777

Offset

0,4

Comments

a(n) = A003108(A000290(n)).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

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

a(n) = [x^(n^2)] Product_{k>=1} 1/(1 - x^(k^3)). - Ilya Gutkovskiy, Jun 05 2017

Example

n=5: number of partitions of 25 into parts of {1, 8}:
a(5) = #{8+8+8+1, 8+8+9x1, 8+17x1, 25x1} = 4;
n=6: number of partitions of 36 into parts of {1, 8, 27}:
a(6) = #{27+8+1, 27+9x1, 4x8+4x1, 3x8+12x1, 8+8+20x1, 8+28x1, 36x1} = 7;
n=7: number of partitions of 49 into parts of {1, 8, 27}:
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.

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +`if`(i^3>n, 0, b(n-i^3, i)))
    end:
a:= n-> b(n^2, iroot(n^2, 3)):
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 08 2012

Mathematica

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 *)

Prog

(Haskell) a218495 = p (tail a000578_list) . (^ 2) where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
(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
(PARI) ok(p)=for(i=1, #p, if(!ispower(p[i], 3), return(0))); 1
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

Crossrefs

Cf. A000578, A218494.

Sequence in context: A082766 A119016 A082958 * A166012 A060166 A053634

Adjacent sequences: A218492 A218493 A218494 * A218496 A218497 A218498

Keyword

nonn

Author

Reinhard Zumkeller, Oct 31 2012

Extensions

More terms from Alois P. Heinz, Nov 08 2012

Status

approved