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%I A000144 #52 Nov 01 2019 11:32:24
%S A000144 1,20,180,960,3380,8424,16320,28800,52020,88660,129064,175680,262080,
%T A000144 386920,489600,600960,840500,1137960,1330420,1563840,2050344,2611200,
%U A000144 2986560,3358080,4194240,5318268,5878440,6299520,7862400,9619560
%N A000144 Number of ways of writing n as a sum of 10 squares.
%D A000144 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
%D A000144 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.
%D A000144 G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135 section 9.3. MR0106147 (21 #4881)
%H A000144 T. D. Noe, <a href="/A000144/b000144.txt">Table of n, a(n) for n = 0..10000</a>
%H A000144 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 A000144 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 A000144 J. Liouville, <a href="http://sites.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1866_2_11_A1_0">Nombre des représentations d’un entier quelconque sous la forme d’une somme de dix carrés</a>, Journal de mathématiques pures et appliquées 2e série, tome 11 (1866), p. 1-8.
%H A000144 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A000144 Euler transform of period 4 sequence [ 20, -30, 20, -10, ...]. - _Michael Somos_, Sep 12 2005
%F A000144 Expansion of eta(q^2)^50 / (eta(q) * eta(q^4))^20 in powers of q. - _Michael Somos_, Sep 12 2005
%F A000144 a(n) = 4/5 * (A050456(n) + 16*A050468(n) + 8*A030212(n)) if n>0. - _Michael Somos_, Sep 12 2005
%F A000144 a(n) = (20/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, May 27 2017
%e A000144 G.f. = 1 + 20*x + 180*x^2 + 960*x^3 + 3380*x^4 + 8424*x^5 + 16320*x^6 + ...
%p A000144 (sum(x^(m^2),m=-10..10))^10;
%p A000144 # Alternative:
%p A000144 A000144list := proc(len) series(JacobiTheta3(0, x)^10, x, len+1);
%p A000144 seq(coeff(%, x, j), j=0..len-1) end: A000144list(30); # _Peter Luschny_, Oct 02 2018
%t A000144 Table[SquaresR[10, n], {n, 0, 30}] (* _Ray Chandler_, Jun 29 2008; updated by _T. D. Noe_, Jan 23 2012 *)
%t A000144 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^10, {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)
%t A000144 nmax = 50; CoefficientList[Series[Product[(1 - x^k)^10 * (1 + x^k)^30 / (1 + x^(2*k))^20, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 24 2017 *)
%o A000144 (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^10, n))}; /* _Michael Somos_, Sep 12 2005 */
%o A000144 (Sage)
%o A000144 Q = DiagonalQuadraticForm(ZZ, [1]*10)
%o A000144 Q.representation_number_list(37) # _Peter Luschny_, Jun 20 2014
%Y A000144 Row d=10 of A122141 and of A319574, 10th column of A286815.
%K A000144 nonn,easy
%O A000144 0,2
%A A000144 _N. J. A. Sloane_
%E A000144 Extended by _Ray Chandler_, Nov 28 2006

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