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%I #10 Mar 12 2021 22:24:46
%S 1,4,4,0,4,8,0,0,4,0,-8,-16,0,-8,-32,0,4,-8,0,16,56,0,16,96,0,-4,24,0,
%T -32,-152,0,-32,-252,0,8,-64,0,56,368,0,56,600,0,-16,144,0,-96,-832,0,
%U -92,-1316,0,24,-312,0,160,1760,0,152,2736,0,-40,640,0,-252,-3536,0,-240,-5432,0,64,-1248,0,392
%N Expansion of (phi(q) / phi(q^9))^2 in powers of q where phi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A164613/b164613.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of (eta(q^2)^5 * eta(q^9)^2 * eta(q^36)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^18)^5))^2 in powers of q.
%F Euler transform of period 36 sequence [ 4, -6, 4, -2, 4, -6, 4, -2, 0, -6, 4, -2, 4, -6, 4, -2, 4, 0, 4, -2, 4, -6, 4, -2, 4, -6, 0, -2, 4, -6, 4, -2, 4, -6, 4, 0, ...].
%F a(3*n) = 0 unless n=0. a(3*n + 1) = 4 * A128111(n). a(3*n + 2) = 4 * A164614(n).
%F Convolution square of A139380.
%e G.f. = 1 + 4*q + 4*q^2 + 4*q^4 + 8*q^5 + 4*q^8 - 8*q^10 - 16*q^11 - 8*q^13 + ...
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^9])^2, {q, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 * eta(x^9 + A)^2 * eta(x^36 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^18 + A)^5))^2, n))};
%Y Cf. A128111, A139380, A164614.
%K sign
%O 0,2
%A _Michael Somos_, Aug 17 2009
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