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A321179
a(n) = [x^(n^2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.
1
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.
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( 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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 29 2018
STATUS
approved