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A261321 Expansion of (phi(q) / phi(q^3))^2 in powers of q where phi() is a Ramanujan theta function.

%I A261321 #14 Feb 16 2025 08:33:26
%S A261321 1,4,4,-4,-12,-8,12,32,20,-28,-72,-48,60,152,96,-120,-300,-184,228,
%T A261321 560,344,-416,-1008,-608,732,1756,1048,-1252,-2976,-1768,2088,4928,
%U A261321 2900,-3408,-7992,-4672,5460,12728,7408,-8600,-19944,-11544,13344,30800,17744,-20424
%N A261321 Expansion of (phi(q) / phi(q^3))^2 in powers of q where phi() is a Ramanujan theta function.
%C A261321 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A261321 The generating function is associated with a modular equation of degree 3 and is the multiplier denoted by "m". - _Michael Somos_, Nov 01 2017
%D A261321 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 230 Entry 5(iii), g.f. denoted by multiplier m.
%H A261321 G. C. Greubel, <a href="/A261321/b261321.txt">Table of n, a(n) for n = 0..1000</a>
%H A261321 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261321 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261321 Expansion of eta(q^2)^10 * eta(q^3)^4 * eta(q^12)^4 / (eta(q)^4 * eta(q^4)^4 * eta(q^6)^10) in powers of q.
%F A261321 G.f.: (Sum_{k in Z} x^k^2) / (Sum_{k in Z} x^(3*k^2))^2.
%F A261321 a(n) = -(1)^n * A217771(n). a(n) = 4 * A187153(n) = 4 * A213265(n) unless n=0.
%F A261321 a(2*n) = 4 * A123633(n) = 4 * A128636(n) unless n=0. a(3*n) = -4 * A228447(n) unless n=0.
%F A261321 Convolution inverse is A261320. Convolution square of A139137.
%e A261321 G.f. = 1 + 4*x + 4*x^2 - 4*x^3 - 12*x^4 - 8*x^5 + 12*x^6 + 32*x^7 + ...
%t A261321 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^3])^2, {q, 0, n}];
%o A261321 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^3 + A)^4 * eta(x^12 + A)^4 / (eta(x + A)^4 * eta(x^4 + A)^4 * eta(x^6 + A)^10), n))};
%Y A261321 Cf. A123633, A128636, A139137, A187153, A213265, A217771, A228447, A261320.
%K A261321 sign
%O A261321 0,2
%A A261321 _Michael Somos_, Aug 14 2015