A302860 a(n) = [x^n] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

1, 3, 9, 27, 89, 333, 1341, 5449, 21697, 84663, 327829, 1275739, 5020457, 19964623, 79883141, 320317827, 1284656385, 5152761033, 20686311261, 83182322509, 335110196569, 1352277390001, 5463873556381, 22097867887045, 89441286136465, 362277846495883, 1468465431530457

Offset: 0

Author

Ilya Gutkovskiy, Apr 14 2018

Keywords

• nonn

Comments

a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius sqrt(n).

Links

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

Crossrefs

Main diagonal of A122510.

Cf. A000122, A001650, A046895, A057655, A066535, A117609, A302861, A066536, A166952.

Programs

Mathematica

Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n}], {n, 0, 26}]

Formula

a(n) = A122510(n,n).

a(n) ~ c / (sqrt(n) * r^n), where r = 0.241970723224463308846762732757915397312... (= radius of convergence A166952) and c = 0.716940866073606328... - Vaclav Kotesovec, Apr 14 2018