%I #42 Oct 02 2018 05:07:05

%S 1,32,480,4480,29152,140736,525952,1580800,3994080,8945824,18626112,

%T 36714624,67978880,118156480,197120256,321692928,509145568,772845120,

%U 1143441760,1681379200,2428524096,3392205824,4658843520,6411152640

%N Number of ways of writing n as a sum of 16 squares.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.

%D E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 107.

%H T. D. Noe, <a href="/A000152/b000152.txt">Table of n, a(n) for n = 0..10000</a>

%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 G.f.: theta_3(0,q)^16, where theta_3 is the 3rd Jacobi theta function. - _Ilya Gutkovskiy_, Jan 13 2017

%F a(n) = (32/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))^16;

%p # Alternative:

%p A000152list := proc(len) series(JacobiTheta3(0, x)^16, x, len+1);

%p seq(coeff(%, x, j), j=0..len-1) end: A000152list(24); # _Peter Luschny_, Oct 02 2018

%t Table[SquaresR[16, n], {n, 0, 23}] (* _Ray Chandler_, Nov 28 2006 *)

%t CoefficientList[EllipticTheta[3, 0, x]^16 + O[x]^24, x] (* _Jean-François Alcover_, Jul 06 2017 *)

%o (PARI) first(n)=my(x='x); x+=O(x^(n+1)); Vec((2*sum(k=1,sqrtint(n),x^k^2) + 1)^16) \\ _Charles R Greathouse IV_, Jul 29 2016

%Y Row d=16 of A122141 and of A319574, 16th column of A286815.

%Y Cf. A022047(n) = A000152(2*n).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E Extended by _Ray Chandler_, Nov 28 2006