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A116597 Expansion of theta_3(q) * theta_4(q^4)^2 in powers of q.

%I A116597 #16 Feb 16 2025 08:33:00
%S A116597 1,2,0,0,-2,-8,0,0,-4,10,0,0,8,-8,0,0,6,16,0,0,-8,-16,0,0,-8,10,0,0,0,
%T A116597 -24,0,0,12,16,0,0,-10,-8,0,0,-8,32,0,0,24,-24,0,0,8,18,0,0,-8,-24,0,
%U A116597 0,-16,16,0,0,0,-24,0,0,6,32,0,0,-16,-32,0,0,-12,16,0,0,24,-32,0,0,24,34,0,0,-16,-16,0,0,-8,48
%N A116597 Expansion of theta_3(q) * theta_4(q^4)^2 in powers of q.
%C A116597 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A116597 G. C. Greubel, <a href="/A116597/b116597.txt">Table of n, a(n) for n = 0..1000</a>
%H A116597 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A116597 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A116597 Expansion of phi(q) * phi(-q^4)^2 in powers of q where phi() is a Ramanujan theta function.
%F A116597 Expansion of eta(q^2)^5 * (eta(q^4) / (eta(q) * eta(q^8)))^2 in powers of q.
%F A116597 Euler transform of period 8 sequence [ 2, -3, 2, -5, 2, -3, 2, -3, ...].
%F A116597 G.f.: theta_3(q) * theta_4(q^4)^2 = Product_{k>0} (1 - x^(2*k))^3 *((1 + x^k) / (1 + x^(4*k)))^2.
%F A116597 a(4*n + 2) = a(4*n + 3) = 0. a(n) = A080963(4*n). a(4*n) = A212885(n). a(4*n + 1) = (-1)^n * A005876(n).
%F A116597 a(3*n + 1) = 2 * A257536(n). - _Michael Somos_, Apr 28 2015
%e A116597 G.f. = 1 + 2*q - 2*q^4 - 8*q^5 - 4*q^8 + 10*q^9 + 8*q^12 - 8*q^13 + 6*q^16 + 16*q^17 + ...
%t A116597 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^4]^2, {q, 0, n}]; (* _Michael Somos_, Apr 28 2015 *)
%o A116597 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * (eta(x^4 + A) / (eta(x + A) * eta(x^8 + A)))^2, n))};
%Y A116597 Cf. A005876, A080963, A212885, A257536.
%K A116597 sign
%O A116597 0,2
%A A116597 _Michael Somos_, Feb 18 2006