A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function.

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 33, 29, 9, 1, 1, 11, 89, 123, 49, 11, 1, 1, 13, 221, 425, 257, 81, 13, 1, 1, 15, 485, 1343, 1281, 515, 113, 15, 1, 1, 17, 953, 4197, 5913, 3121, 925, 149, 17, 1, 1, 19, 1713, 12435, 23793, 16875, 6577, 1419, 197, 19, 1, 1, 21, 2869, 33809, 88273, 84769, 42205, 11833, 2109, 253, 21, 1

Offset: 0

Author

Ilya Gutkovskiy, Apr 17 2018

Keywords

• nonn
• tabl

Comments

A(n,k) is the number of integer lattice points inside the k-dimensional hypersphere of radius n.

Examples

			Square array begins:
  1,   1,   1,    1,     1,      1,  ...
  1,   3,   5,    7,     9,     11,  ...
  1,   5,  13,   33,    89,    221,  ...
  1,   7,  29,  123,   425,   1343,  ...
  1,   9,  49,  257,  1281,   5913,  ...
  1,  11,  81,  515,  3121,  16875,  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions
• Index entries for sequences related to sums of squares

Crossrefs

Columns k=0..10 give A000012, A005408, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416.

Rows k=0..10 give A000012, A005408, A055426, A055427, A055428, A055429, A055430, A055431, A055432, A055433, A055434.

Main diagonal gives A302861.

Cf. A000122, A122510, A302996, A302998.

Programs

Mathematica

Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k/(1 - x), {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

PARI

T(n,k)={if(k==0, 1, polcoef(((1 + 2*sum(j=1, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ Andrew Howroyd, Sep 14 2019

Formula

A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j=-infinity..infinity} x^(j^2))^k.