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A218494 Number of partitions of n^3 into squares.

%I A218494 #20 Jan 29 2018 21:11:45
%S A218494 1,1,3,21,220,2846,41621,670568,11570877,210605770,3998468431,
%T A218494 78556582448,1587757499592,32866068149376,694307793698105,
%U A218494 14927522659021682,325895131806047690,7211436102222542901,161493494674514291108,3655277488432342084426
%N A218494 Number of partitions of n^3 into squares.
%H A218494 Alois P. Heinz, <a href="/A218494/b218494.txt">Table of n, a(n) for n = 0..80</a>
%F A218494 a(n) = A001156(A000578(n)).
%F A218494 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
%F A218494 a(n) = [x^(n^3)] Product_{k>=1} 1/(1 - x^(k^2)). - _Ilya Gutkovskiy_, Jan 29 2018
%e A218494 n=2: number of partitions of 8 into parts of {1, 4}:
%e A218494 a(2) = #{4+4, 4+1+1+1+1, 8x1} = 3;
%e A218494 n=3: number of partitions of 27 into parts of  {1, 4, 9, 16, 25}:
%e A218494 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.
%p A218494 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A218494        b(n, i-1)+`if`(i^2>n, 0, b(n-i^2, i)))
%p A218494     end:
%p A218494 a:= n-> b(n^3, isqrt(n^3)):
%p A218494 seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 08 2012
%t A218494 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_ *)
%o A218494 (Haskell)
%o A218494 a218494 = p (tail a000290_list) . (^ 3) where
%o A218494    p _          0 = 1
%o A218494    p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%Y A218494 Cf. A000290, A001156, A218495.
%K A218494 nonn
%O A218494 0,3
%A A218494 _Reinhard Zumkeller_, Oct 31 2012
%E A218494 Extended beyond a(7) by _Alois P. Heinz_, Nov 08 2012