A321179 a(n) = [x^(n^2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.

1, 2, 2, 14, 44, 174, 988, 4314, 20780, 126320, 692328, 3836166, 23160914, 135752866, 803203484, 4902966108, 29745996950, 181712320506, 1124481497694, 6965802854354, 43360326335154, 271658784580760, 1706393926177980, 10757142052998054, 68081390206251952, 432001821971576352

Offset

0,2

Comments

Also the number of integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n^2.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (first 91 terms from Seiichi Manyama)

Formula

a(n) ~ c * d^n / n^(7/4), where d = 6.8137220913147... and c = 0.178176349247... - Vaclav Kotesovec, Oct 30 2018

Example

Solutions (a_1, a_2, a_3) to the equation a_1^2 + 2*a_2^2 + 3*a_3^2 = 9.
------------------------------------------------------------------------
( 1,  2,  0), ( 1, -2,  0),
(-1,  2,  0), (-1, -2,  0),
( 2,  1,  1), ( 2,  1, -1),
( 2, -1,  1), ( 2, -1, -1),
(-2,  1,  1), (-2,  1, -1),
(-2, -1,  1), (-2, -1, -1),
( 3,  0,  0), (-3,  0,  0).

Mathematica

nmax = 20; Table[SeriesCoefficient[Product[EllipticTheta[3, 0, x^k], {k, 1, n}], {x, 0, n^2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)

Prog

(PARI) {a(n) = polcoeff(prod(i=1, n, 1+2*sum(j=1, sqrtint(n^2\i), x^(i*j^2)+x*O(x^(n^2)))), n^2)}

Crossrefs

Cf. A000122, A000290, A320067, A320931, A321139.

Sequence in context: A264508 A194689 A277556 * A345370 A166114 A366364

Adjacent sequences: A321176 A321177 A321178 * A321180 A321181 A321182

Keyword

nonn

Author

Seiichi Manyama, Oct 29 2018

Status

approved