This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A319574 #30 Oct 19 2022 11:06:37
%S A319574 1,0,1,0,2,1,0,0,4,1,0,0,4,6,1,0,2,0,12,8,1,0,0,4,8,24,10,1,0,0,8,6,
%T A319574 32,40,12,1,0,0,0,24,24,80,60,14,1,0,0,0,24,48,90,160,84,16,1,0,2,4,0,
%U A319574 96,112,252,280,112,18,1,0,0,4,12,64,240,312,574,448,144,20,1
%N A319574 A(n, k) = [x^k] JacobiTheta3(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.
%C A319574 Number of ways of writing k as a sum of n squares.
%D A319574 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
%D A319574 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954.
%D A319574 J. Carlos Moreno and Samuel S. Wagstaff Jr., Sums Of Squares Of Integers, Chapman & Hall/CRC, (2006).
%H A319574 Seiichi Manyama, <a href="/A319574/b319574.txt">Descending antidiagonals n = 0..139, flattened</a>
%H A319574 L. Carlitz, <a href="http://dx.doi.org/10.1090/S0002-9939-1957-0084520-2">Note on sums of four and six squares</a>, Proc. Amer. Math. Soc. 8 (1957), 120-124.
%H A319574 S. H. Chan, <a href="http://www.jstor.org/stable/4145192">An elementary proof of Jacobi's six squares theorem</a>, Amer. Math. Monthly, 111 (2004), 806-811.
%H A319574 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 A319574 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 A319574 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 A319574 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%e A319574 [ 0] 1, 0, 0, 0, 0, 0, 0 0, 0, 0, ... A000007
%e A319574 [ 1] 1, 2, 0, 0, 2, 0, 0, 0, 0, 2, ... A000122
%e A319574 [ 2] 1, 4, 4, 0, 4, 8, 0, 0, 4, 4, ... A004018
%e A319574 [ 3] 1, 6, 12, 8, 6, 24, 24, 0, 12, 30, ... A005875
%e A319574 [ 4] 1, 8, 24, 32, 24, 48, 96, 64, 24, 104, ... A000118
%e A319574 [ 5] 1, 10, 40, 80, 90, 112, 240, 320, 200, 250, ... A000132
%e A319574 [ 6] 1, 12, 60, 160, 252, 312, 544, 960, 1020, 876, ... A000141
%e A319574 [ 7] 1, 14, 84, 280, 574, 840, 1288, 2368, 3444, 3542, ... A008451
%e A319574 [ 8] 1, 16, 112, 448, 1136, 2016, 3136, 5504, 9328, 12112, ... A000143
%e A319574 [ 9] 1, 18, 144, 672, 2034, 4320, 7392, 12672, 22608, 34802, ... A008452
%e A319574 [10] 1, 20, 180, 960, 3380, 8424, 16320, 28800, 52020, 88660, ... A000144
%e A319574 A005843, v, A130809, v, A319576, v , ... diagonal: A066535
%e A319574 A046092, A319575, A319577, ...
%p A319574 A319574row := proc(n, len) series(JacobiTheta3(0, x)^n, x, len+1);
%p A319574 [seq(coeff(%, x, j), j=0..len-1)] end:
%p A319574 seq(print([n], A319574row(n, 10)), n=0..10);
%p A319574 # Alternative, uses function PMatrix from A357368.
%p A319574 PMatrix(10, n -> A000122(n-1)); # _Peter Luschny_, Oct 19 2022
%t A319574 A[n_, k_] := If[n == k == 0, 1, SquaresR[n, k]];
%t A319574 Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Nov 03 2018 *)
%o A319574 (Sage)
%o A319574 for n in (0..10):
%o A319574 Q = DiagonalQuadraticForm(ZZ, [1]*n)
%o A319574 print(Q.theta_series(10).list())
%Y A319574 Variant starting with row 1 is A122141, transpose of A286815.
%K A319574 nonn,tabl
%O A319574 0,5
%A A319574 _Peter Luschny_, Oct 01 2018