A218494 Number of partitions of n^3 into squares.

1, 1, 3, 21, 220, 2846, 41621, 670568, 11570877, 210605770, 3998468431, 78556582448, 1587757499592, 32866068149376, 694307793698105, 14927522659021682, 325895131806047690, 7211436102222542901, 161493494674514291108, 3655277488432342084426

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..80

Formula

a(n) = A001156(A000578(n)).

a(n) ~ 3^(-1/2) * (4*Pi)^(-7/6) * Zeta(3/2)^(2/3) * n^(-7/2) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n) [after Hardy & Ramanujan]. - Vaclav Kotesovec, Apr 10 2017

a(n) = [x^(n^3)] Product_{k>=1} 1/(1 - x^(k^2)). - Ilya Gutkovskiy, Jan 29 2018

Example

n=2: number of partitions of 8 into parts of {1, 4}:
a(2) = #{4+4, 4+1+1+1+1, 8x1} = 3;
n=3: number of partitions of 27 into parts of  {1, 4, 9, 16, 25}:
a(3) = #{25+1+1, 16+9+1+1, 16+4+4+1, 16+4+5x1, 16+9x1, 9+9+9, 9+9+4+4+1, 9+9+4+5x1, 9+9+9x1, 9+4x4+1+1, 9+3x4+6x1, 9+4+4+10x1, 9+4+14x1, 9+18x1, 6x4+3x1, 5x4+7x1, 4x4+11x1, 3x4+15x1, 4+4+19x1, 4+23x1, 27x1} = 21.

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
       b(n, i-1)+`if`(i^2>n, 0, b(n-i^2, i)))
    end:
a:= n-> b(n^3, isqrt(n^3)):
seq(a(n), n=0..20);  # 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^2>n, 0, b[n - i^2, i]]]; a[n_] := b[n^3, Sqrt[n^3] // Floor]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

Prog

(Haskell)
a218494 = p (tail a000290_list) . (^ 3) where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

Crossrefs

Cf. A000290, A001156, A218495.

Sequence in context: A236963 A107716 A032033 * A099121 A107864 A267657

Adjacent sequences: A218491 A218492 A218493 * A218495 A218496 A218497

Keyword

nonn

Author

Reinhard Zumkeller, Oct 31 2012

Extensions

Extended beyond a(7) by Alois P. Heinz, Nov 08 2012

Status

approved