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%I #16 Mar 12 2021 22:24:46
%S 1,1,-1,0,1,0,1,1,0,0,-1,0,1,-1,0,0,0,0,-1,1,0,0,2,0,-1,0,0,0,-2,0,0,
%T -3,0,0,-1,0,1,4,0,0,4,0,2,1,0,0,-4,0,0,-6,0,0,-1,0,-2,5,0,0,8,0,-3,1,
%U 0,0,-8,0,-1,-10,0,0,-2,0,4,11,0,0,14,0,4,4,0,0,-14,0,1,-19,0,0,-4,0,-4,17,0,0,24,0,-6,4,0,0,-23,0
%N Expansion of eta(q^2)^3 * eta(q^9) * eta(q^12)^4 / (eta(q) * eta(q^4)^2 * eta(q^6)^2 * eta(q^18) * eta(q^36)^2) in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A182032/b182032.txt">Table of n, a(n) for n = -2..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 q^(-2) * chi(q) * chi(-q^2) * psi(q^6)^2 / (psi(q^9) * psi(q^18)) in powers of q where psi(), chi() are Ramanujan theta functions.
%F Euler transform of period 36 sequence [ 1, -2, 1, 0, 1, 0, 1, 0, 0, -2, 1, -2, 1, -2, 1, 0, 1, 0, 1, 0, 1, -2, 1, -2, 1, -2, 0, 0, 1, 0, 1, 0, 1, -2, 1, 0, ...].
%F a(6*n) = 0 unless n=0. a(6*n + 1) = a(6*n + 3) = 0.
%F A062242(n) = a(3*n - 1). A132179(n) = a(6*n - 1). A062242(n) = a(6*n - 2). A092848(n) = a(6*n + 2).
%e q^-2 + q^-1 - 1 + q^2 + q^4 + q^5 - q^8 + q^10 - q^11 - q^16 + q^17 + ...
%t QP = QPochhammer; s = QP[q^2]^3*QP[q^9]*(QP[q^12]^4 / (QP[q]*QP[q^4]^2* QP[q^6]^2*QP[q^18]*QP[q^36]^2)) + O[q]^90; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 24 2015 *)
%o (PARI) {a(n) = local(A); if( n<-2, 0, n+=2; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^9 + A) * eta(x^12 + A)^4 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^2 * eta(x^18 + A) * eta(x^36 + A)^2), n))}
%Y Cf. A062242, A092848, A132179.
%K sign
%O -2,23
%A _Michael Somos_, Apr 07 2012
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