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%I #19 Mar 12 2021 22:24:47
%S 1,3,0,-5,0,0,-7,0,0,-3,0,0,15,0,0,32,0,0,9,0,0,-58,0,0,-96,0,0,-22,0,
%T 0,149,0,0,253,0,0,68,0,0,-372,0,0,-599,0,0,-140,0,0,826,0,0,1317,0,0,
%U 317,0,0,-1768,0,0,-2735,0,0,-632,0,0,3526,0,0,5434
%N Expansion of (1/q) * (f(q) / f(q^9))^3 in powers of q where f() 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="/A227498/b227498.txt">Table of n, a(n) for n = -1..2500</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 -3 * b(-q) / c(-q^3) in powers of q where b(), c() are cubic AGM theta functions.
%F Expansion of ( eta(q^2)^3 * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^18)^3) )^3 in powers of q.
%F Euler transform of period 36 sequence [3, -6, 3, -3, 3, -6, 3, -3, 0, -6, 3, -3, 3, -6, 3, -3, 3, 0, 3, -3, 3, -6, 3, -3, 3, -6, 0, -3, 3, -6, 3, -3, 3, -6, 3, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 27 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A227454.
%F G.f.: (1/x) * (Product_{k>0} (1 - (-x)^k) / (1 - (-x)^(9*k)))^3.
%F a(3*n) = 0 unless n=0. a(3*n + 1) = 0. a(3*n-1) = (-1)^n * A058091(n).
%F Convolution inverse of A227454.
%e G.f. = 1/q + 3 - 5*q^2 - 7*q^5 - 3*q^8 + 15*q^11 + 32*q^14 + 9*q^17 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] / QPochhammer[ -q^9])^3 / q, {q, 0, n}];
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^3 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^3))^3, n))};
%Y Cf. A058091, A227454.
%K sign
%O -1,2
%A _Michael Somos_, Sep 22 2013
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