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

0,5

Comments

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

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

Formula

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

Example

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,  ...

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

Crossrefs

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

Main diagonal gives A232173.

Cf. A000122, A122141, A255212, A286815.

Sequence in context: A246862 A338773 A194686 * A266213 A289522 A361397

Adjacent sequences: A302993 A302994 A302995 * A302997 A302998 A302999

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Apr 17 2018

Status

approved