A319574 A(n, k) = [x^k] JacobiTheta3(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

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

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.

Index entries for sequences related to sums of squares

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);
# Alternative, uses function PMatrix from A357368.
PMatrix(10, n -> A000122(n-1)); # Peter Luschny, Oct 19 2022

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