A302995 a(n) = [x^(n^2)] (theta_3(x) - 1)^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

1, 1, 1, 7, 32, 177, 1269, 9263, 74452, 652710, 6078048, 60447082, 631870024, 6915613084, 79113376037, 941759419159, 11630647314564, 148799595377384, 1966441829785081, 26793749867965515, 375812005722920406, 5416574818546042067, 80123280319100908258, 1214860029446181979357

Offset: 0

Author

Ilya Gutkovskiy, Apr 17 2018

Keywords

• nonn

Comments

a(n) = number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_n)^2 <= n^2.

Links

• Eric Weisstein's World of Mathematics, Jacobi Theta Functions
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000122, A001182, A253663, A298330, A302861, A302863.

Programs

Mathematica

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^n/(2^n (1 - x)), {x, 0, n^2}], {n, 0, 23}]
Join[{1}, Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, 1, n}]^n, {x, 0, n^2}], {n, 23}]]