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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.

%I A321179 #28 Oct 30 2018 03:19:44
%S A321179 1,2,2,14,44,174,988,4314,20780,126320,692328,3836166,23160914,
%T A321179 135752866,803203484,4902966108,29745996950,181712320506,
%U A321179 1124481497694,6965802854354,43360326335154,271658784580760,1706393926177980,10757142052998054,68081390206251952,432001821971576352
%N A321179 a(n) = [x^(n^2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.
%C A321179 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.
%H A321179 Vaclav Kotesovec, <a href="/A321179/b321179.txt">Table of n, a(n) for n = 0..400</a> (first 91 terms from Seiichi Manyama)
%F A321179 a(n) ~ c * d^n / n^(7/4), where d = 6.8137220913147... and c = 0.178176349247... - _Vaclav Kotesovec_, Oct 30 2018
%e A321179 Solutions (a_1, a_2, a_3) to the equation a_1^2 + 2*a_2^2 + 3*a_3^2 = 9.
%e A321179 ------------------------------------------------------------------------
%e A321179 ( 1,  2,  0), ( 1, -2,  0),
%e A321179 (-1,  2,  0), (-1, -2,  0),
%e A321179 ( 2,  1,  1), ( 2,  1, -1),
%e A321179 ( 2, -1,  1), ( 2, -1, -1),
%e A321179 (-2,  1,  1), (-2,  1, -1),
%e A321179 (-2, -1,  1), (-2, -1, -1),
%e A321179 ( 3,  0,  0), (-3,  0,  0).
%t A321179 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 *)
%o A321179 (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)}
%Y A321179 Cf. A000122, A000290, A320067, A320931, A321139.
%K A321179 nonn
%O A321179 0,2
%A A321179 _Seiichi Manyama_, Oct 29 2018