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A037444 Number of partitions of n^2 into squares.

%I A037444 #64 Oct 31 2018 06:40:39
%S A037444 1,1,2,4,8,19,43,98,220,504,1116,2468,5368,11592,24694,52170,108963,
%T A037444 225644,462865,941528,1899244,3801227,7550473,14889455,29159061,
%U A037444 56722410,109637563,210605770,402165159,763549779,1441686280,2707535748,5058654069,9404116777
%N A037444 Number of partitions of n^2 into squares.
%C A037444 Is lim_{n->inf} a(n)^(1/n) > 1? - _Paul D. Hanna_, Aug 20 2002
%C A037444 The limit above is equal to 1 (see formula by Hardy & Ramanujan for A001156). - _Vaclav Kotesovec_, Dec 29 2016
%H A037444 T. D. Noe, Alois P. Heinz and Vaclav Kotesovec, <a href="/A037444/b037444.txt">Table of n, a(n) for n = 0..945</a> (terms n = 0..100 from T. D. Noe, terms n = 101..500 from Alois P. Heinz)
%H A037444 J. Bohman et al., <a href="http://dx.doi.org/10.1007/BF01930983">Partitions in squares</a>, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
%H A037444 H. L. Fisher, <a href="/A027601/a027601.pdf">Letter to N. J. A. Sloane, Mar 16 1989</a>
%H A037444 G. H. Hardy and S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram33.html">Asymptotic formulae in combinatory analysis</a>, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.
%F A037444 a(n) = A001156(n^2) = coefficient of x^(n^2) in the series expansion of Prod_{k>=1} 1/(1 - x^(k^2)).
%F A037444 a(n) ~ 3^(-1/2) * (4*Pi)^(-7/6) * Zeta(3/2)^(2/3) * n^(-7/3) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(2/3)) [Hardy & Ramanujan, 1917, modified from A001156]. - _Vaclav Kotesovec_, Dec 29 2016
%p A037444 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A037444       b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i)))
%p A037444     end:
%p A037444 a:= n-> b(n^2, n):
%p A037444 seq(a(n), n=0..40);  # _Alois P. Heinz_, Apr 15 2013
%t A037444 max=33; se = Series[ Product[1/(1-x^(k^2)), {k, 1, max}], {x, 0, max^2}]; a[n_] := Coefficient[se, x^(n^2)]; a[0] = 1; Table[a[n], {n, 0, max}] (* _Jean-François Alcover_, Oct 18 2011 *)
%o A037444 (Haskell)
%o A037444 a037444 n = p (map (^ 2) [1..]) (n^2) where
%o A037444    p _      0 = 1
%o A037444    p ks'@(k:ks) m | m < k     = 0
%o A037444                   | otherwise = p ks' (m - k) + p ks m
%o A037444 -- _Reinhard Zumkeller_, Aug 14 2011
%Y A037444 Entries with square index in A001156.
%Y A037444 Cf. A072964, A030273, A000041, A000290, A229239, A229468.
%Y A037444 Cf. A003108, A046042.
%Y A037444 Cf. A259792, A259793.
%Y A037444 A row or column of the array in A259799.
%K A037444 nonn,nice,easy
%O A037444 0,3
%A A037444 _Wouter Meeussen_