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A224821 Expansion of theta_4(q)^2 * theta_4(q^3) in powers of q.

%I A224821 #17 Feb 16 2025 08:33:19
%S A224821 1,-4,4,-2,12,-16,0,-8,20,-4,8,-8,10,-32,8,0,28,-24,4,-8,32,-16,16,
%T A224821 -16,0,-28,8,-2,40,-48,8,-8,52,0,8,-16,12,-64,16,-8,40,-24,0,-24,40,
%U A224821 -16,16,-16,26,-28,20,0,64,-80,0,-16,40,-24,24,-8,0,-64,24,-8
%N A224821 Expansion of theta_4(q)^2 * theta_4(q^3) in powers of q.
%C A224821 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A224821 G. C. Greubel, <a href="/A224821/b224821.txt">Table of n, a(n) for n = 0..1000</a>
%H A224821 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A224821 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A224821 Expansion of phi(-q)^2 * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.
%F A224821 Expansion of eta(q)^4 * eta(q^3)^2 / (eta(q^2)^2 * eta(q^6)) in powers of q.
%F A224821 G.f.: theta_4(q)^2 * theta_4(q^3) = (Sum_{k in Z} (-1)^k * x^k^2)^2 * (Sum_{k in Z} (-1)^k * x^(3*k^2)).
%F A224821 a(n) = (-1)^n * A034933(n). a(2*n) = A014458(n). a(9*n) = a(n). a(9*n + 6) = 0.
%e A224821 1 - 4*q + 4*q^2 - 2*q^3 + 12*q^4 - 16*q^5 - 8*q^7 + 20*q^8 - 4*q^9 + ...
%t A224821 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^3], {q, 0 ,n}]
%o A224821 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^3 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))}
%o A224821 (PARI) q='q+O('q^99); Vec(eta(q)^4*eta(q^3)^2/(eta(q^2)^2*eta(q^6))) \\ _Altug Alkan_, Apr 12 2018
%Y A224821 Cf. A014458, A034933.
%K A224821 sign
%O A224821 0,2
%A A224821 _Michael Somos_, Jul 20 2013