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A261320 Expansion of (phi(q^3) / phi(q))^2 in powers of q where phi() is a Ramanujan theta function.

%I A261320 #10 Feb 16 2025 08:33:26
%S A261320 1,-4,12,-28,60,-120,228,-416,732,-1252,2088,-3408,5460,-8600,13344,
%T A261320 -20424,30876,-46152,68268,-100016,145224,-209120,298800,-423840,
%U A261320 597108,-835804,1162824,-1608508,2212896,-3028632,4124664,-5590976,7544604,-10137264,13565016
%N A261320 Expansion of (phi(q^3) / phi(q))^2 in powers of q where phi() is a Ramanujan theta function.
%C A261320 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261320 Vaclav Kotesovec, <a href="/A261320/b261320.txt">Table of n, a(n) for n = 0..2000</a>
%H A261320 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261320 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261320 Expansion of eta(q)^4 * eta(q^4)^4 * eta(q^6)^10 / ( eta(q^2)^10 * eta(q^3)^4 * eta(q^12)^4) in powers of q.
%F A261320 Euler transform of period 12 sequence [ -4, 6, 0, 2, -4, 0, -4, 2, 0, 6, -4, 0, ...].
%F A261320 G.f.: (Sum_{k in Z} x^(3*k^2)) / (Sum_{k in Z} x^k^2)^2.
%F A261320 G.f.: (Product_{k>0} (1 + (-x)^k + x^(2*k)) / (1 - (-x)^k + x^(2*k)))^2.
%F A261320 a(n) = (-1)^n * A186924(n) = A233673(3*n) = A260215(3*n).
%F A261320 Convolution square of A132002.
%F A261320 a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2*3^(5/4)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e A261320 G.f. = 1 - 4*x + 12*x^2 - 28*x^3 + 60*x^4 - 120*x^5 + 228*x^6 + ...
%t A261320 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^3] / EllipticTheta[ 3, 0, q])^2, {q, 0, n}];
%o A261320 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A)^4 * eta(x^6 + A)^10 / (eta(x^2 + A)^10 * eta(x^3 + A)^4 * eta(x^12 + A)^4), n))};
%Y A261320 Cf. A132002, A186924, A260215, A233673.
%K A261320 sign
%O A261320 0,2
%A A261320 _Michael Somos_, Aug 14 2015