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 A319574 A(n, k) = [x^k] JacobiTheta3(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0. 18
 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, 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, 96, 112, 252, 280, 112, 18, 1, 0, 0, 4, 12, 64, 240, 312, 574, 448, 144, 20, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Number of ways of writing k as a sum of n squares. REFERENCES E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954. J. Carlos Moreno and Samuel S. Wagstaff Jr., Sums Of Squares Of Integers, Chapman & Hall/CRC, (2006). LINKS Seiichi Manyama, Descending antidiagonals n = 0..139, flattened L. Carlitz, Note on sums of four and six squares, Proc. Amer. Math. Soc. 8 (1957), 120-124. S. H. Chan, An elementary proof of Jacobi's six squares theorem, Amer. Math. Monthly, 111 (2004), 806-811. H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004. Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032. S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149. EXAMPLE [ 0] 1,  0,    0,    0,     0,     0,     0      0,     0,     0, ... A000007 [ 1] 1,  2,    0,    0,     2,     0,     0,     0,     0,     2, ... A000122 [ 2] 1,  4,    4,    0,     4,     8,     0,     0,     4,     4, ... A004018 [ 3] 1,  6,   12,    8,     6,    24,    24,     0,    12,    30, ... A005875 [ 4] 1,  8,   24,   32,    24,    48,    96,    64,    24,   104, ... A000118 [ 5] 1, 10,   40,   80,    90,   112,   240,   320,   200,   250, ... A000132 [ 6] 1, 12,   60,  160,   252,   312,   544,   960,  1020,   876, ... A000141 [ 7] 1, 14,   84,  280,   574,   840,  1288,  2368,  3444,  3542, ... A008451 [ 8] 1, 16,  112,  448,  1136,  2016,  3136,  5504,  9328, 12112, ... A000143 [ 9] 1, 18,  144,  672,  2034,  4320,  7392, 12672, 22608, 34802, ... A008452 [10] 1, 20,  180,  960,  3380,  8424, 16320, 28800, 52020, 88660, ... A000144    A005843,   v, A130809,  v,  A319576,  v ,   ...      diagonal: A066535            A046092,    A319575,       A319577,     ... MAPLE A319574row := proc(n, len) series(JacobiTheta3(0, x)^n, x, len+1); [seq(coeff(%, x, j), j=0..len-1)] end: seq(print([n], A319574row(n, 10)), n=0..10); MATHEMATICA A[n_, k_] := If[n == k == 0, 1, SquaresR[n, k]]; Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2018 *) PROG (Sage) for n in (0..10):     Q = DiagonalQuadraticForm(ZZ, [1]*n)     print(Q.theta_series(10).list()) CROSSREFS Variant starting with row 1 is A122141, transpose of A286815. Sequence in context: A204387 A110509 A113953 * A204040 A325773 A220779 Adjacent sequences:  A319571 A319572 A319573 * A319575 A319576 A319577 KEYWORD nonn,tabl AUTHOR Peter Luschny, Oct 01 2018 STATUS approved

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Last modified April 20 06:59 EDT 2021. Contains 343125 sequences. (Running on oeis4.)