A259792 Number of partitions of n^3 into cubes.

1, 1, 2, 5, 17, 62, 258, 1050, 4365, 18012, 73945, 301073, 1214876, 4852899, 19187598, 75070201, 290659230, 1113785613, 4224773811, 15866483556, 59011553910, 217410395916, 793635925091, 2871246090593, 10297627606547, 36620869115355, 129166280330900

Offset: 0

Author

N. J. A. Sloane, Jul 06 2015

Keywords

• nonn

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..173 (terms 0..120 from Alois P. Heinz)
• H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
• G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.

Crossrefs

A row of the array in A259799.

Cf. A279329.

Cf. A001156, A003108, A046042.

Cf. A037444, A259793.

Programs

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^3, n):
seq(a(n), n=0..26);  # Alois P. Heinz, Jul 10 2015

Mathematica

$RecursionLimit = 1000; 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^3, n]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

Formula

a(n) = [x^(n^3)] Product_{j>=1} 1/(1-x^(j^3)). - Alois P. Heinz, Jul 10 2015

a(n) = A003108(n^3). - Vaclav Kotesovec, Aug 19 2015

a(n) ~ exp(4 * (Gamma(1/3)*Zeta(4/3))^(3/4) * n^(3/4) / 3^(3/2)) * (Gamma(1/3)*Zeta(4/3))^(3/4) / (24*Pi^2*n^(15/4)) [after Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016

Extensions

More term from Alois P. Heinz, Jul 10 2015