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A218494 Number of partitions of n^3 into squares. 7
1, 1, 3, 21, 220, 2846, 41621, 670568, 11570877, 210605770, 3998468431, 78556582448, 1587757499592, 32866068149376, 694307793698105, 14927522659021682, 325895131806047690, 7211436102222542901, 161493494674514291108, 3655277488432342084426 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified March 28 17:51 EDT 2020. Contains 333103 sequences. (Running on oeis4.)