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%I #63 Oct 02 2018 05:03:10
%S 1,48,1104,16192,170064,1362336,8662720,44981376,195082320,721175536,
%T 2319457632,6631997376,17231109824,41469483552,93703589760,
%U 200343312768,407488018512,793229226336,1487286966928,2697825744960,4744779429216
%N Number of ways of writing n as a sum of 24 squares.
%C The Carlitz paper has the wrong formula on p. 505, eq. (3). The factor in front of tau(n/2) should be -2^16 (not -2^12). The mistake appeared in the previous equation derived from eq. (2) where theta_3^(24) * 256*k^4*k'^4 was replaced by 2^8*g(q^2) which produces the factor 2^8*256 = 2^16. (One should subtract on p. 504 the second equation in the middle from the negative of the first equation. There is also a sign mistake in the sum term of the third equation from the bottom.) - _Wolfdieter Lang_, Sep 24 2016
%D Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 195, eq. (15.1).
%D E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 107.
%D G. H. Hardy, Ramanujan, 1940, Cambridge, reprinted with additional corrections and comments by AMS Chelsea Publishing, 1999, 2002, Providence, Rhode Island, ch. IX., pp. 153-155.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.
%H T. D. Noe, <a href="/A000156/b000156.txt">Table of n, a(n) for n = 0..10000</a>
%H L. Carlitz, <a href="http://www.sciencedirect.com/science/article/pii/S1385725855500680">On the representation of an integer as the sum of twenty-four squares</a>, Indagationes Mathematicae (Proceedings), 58 (1955) 504-506.
%H H. H. Chan and C. Krattenthaler, <a href="http://arXiv.org/abs/math.NT/0407061">Recent progress in the study of representations of integers as sums of squares</a>, arXiv:math/0407061 [math.NT], 2004.
%H Shi-Chao Chen, <a href="http://dx.doi.org/10.1016/j.jnt.2010.01.011">Congruences for rs(n)</a>, Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
%H S. C. Milne, <a href="http://dx.doi.org/10.1023/A:1014865816981">Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions</a>, Ramanujan J., 6 (2002), 7-149.
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F From _Wolfdieter Lang_, Sep 24 2016: (Start)
%F For n >= 1: a(n) = (16*sigma^*_{11} - 128*(512*tau(n/2) + (-1)^n*259*tau(n)))/691, with sigma^*_{11} = sigma_{11}^{e}(n) - sigma_{11}^{o}(n) if n even and sigma_{11}(n) otherwise. Here sigma_{11}(n) = A013959(n) and 0 if n is not an integer, sigma_{11}^{e}(n) and sigma_{11}^{o}(n) are the sums of the 11th power of the odd and even positive divisors of n, respectively. Ramanujan's tau(n) = A000594(n) and 0 if n is not an integer. This is from Hardy, ch. IX., p. 155, eqs. (9.17.1) and (9.17.2), and p.142 for the definition of sigma^*_{nu}(n). See also the Ash and Gross reference.
%F Another version, see the corrected Carlitz reference:
%F a(n) = (2^4*(sigma_{11}(n)- 2*sigma_{11}(n/2) + 2^{12}*sigma_{11}(n/4)) - 2^7*259*(-1)^n*tau(n) - 2^16*tau(n/2))/691, n >= 1.
%F (End)
%F a(n) = (48/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, May 27 2017
%p (sum(x^(m^2),m=-10..10))^24; seq(coeff(%,x,n), n=0..30);
%p # Alternative:
%p A000156list := proc(len) series(JacobiTheta3(0, x)^24, x, len+1);
%p seq(coeff(%, x, j), j=0..len-1) end: A000156list(21); # _Peter Luschny_, Oct 02 2018
%t Table[SquaresR[24, n], {n, 0, 20}] (* _Ray Chandler_, Nov 28 2006 *)
%o (PARI) first(n)=my(x='x); x+=O(x^(n+1)); Vec((2*sum(k=1,sqrtint(n),x^k^2) + 1)^24) \\ _Charles R Greathouse IV_, Jul 29 2016
%Y Row d=24 of A122141 and of A319574, 24th column of A286815.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Extended by _Ray Chandler_, Nov 28 2006
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