A302996 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k, where theta_3() is the Jacobi theta function.

1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 4, 2, 0, 1, 8, 6, 4, 2, 0, 1, 10, 24, 30, 4, 2, 0, 1, 12, 90, 104, 6, 12, 2, 0, 1, 14, 252, 250, 24, 30, 4, 2, 0, 1, 16, 574, 876, 730, 248, 30, 4, 2, 0, 1, 18, 1136, 3542, 4092, 1210, 312, 54, 4, 2, 0, 1, 20, 2034, 12112, 18494, 7812, 2250, 456, 6, 4, 2, 0

Offset: 0

Author

Ilya Gutkovskiy, Apr 17 2018

Keywords

• nonn
• tabl

Comments

A(n,k) is the number of ordered ways of writing n^2 as a sum of k squares.

Examples

			Square array begins:
  1,  1,   1,   1,    1,     1,  ...
  0,  2,   4,   6,    8,    10,  ...
  0,  2,   4,   6,   24,    90,  ...
  0,  2,   4,  30,  104,   250,  ...
  0,  2,   4,   6,   24,   730,  ...
  0,  2,  12,  30,  248,  1210,  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..200, flattened
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions
• Index entries for sequences related to sums of squares

Crossrefs

Columns k=0..4,7 give A000007, A040000, A046109, A016725, A267326, A361695.

Main diagonal gives A232173.

Cf. A000122, A122141, A255212, A286815.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
      b(n, t-1)+2*add(b(n-j^2, t-1), j=1..isqrt(n))))
    end:
A:= (n, k)-> b(n^2, k):
seq(seq(A(n,d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 10 2023

Mathematica

Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

A(n,k) = [x^(n^2)] (Sum_{j=-infinity..infinity} x^(j^2))^k.