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Expansion of phi(-q^2) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.
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%I #11 Mar 12 2021 22:24:48

%S 1,0,-2,2,0,-4,4,0,-6,8,0,-12,14,0,-20,24,0,-32,38,0,-52,60,0,-80,92,

%T 0,-120,138,0,-180,204,0,-262,296,0,-376,424,0,-536,600,0,-752,840,0,

%U -1044,1164,0,-1440,1598,0,-1966,2176,0,-2660,2940,0,-3580,3944,0

%N Expansion of phi(-q^2) / phi(-q^3) 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="/A262967/b262967.txt">Table of n, a(n) for n = 0..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 eta(q^2)^2 * eta(q^6) / (eta(q^3)^2 * eta(q^4)) in powers of q.

%F Euler transform of period 12 sequence [0, -2, 2, -1, 0, -1, 0, -1, 2, -2, 0, 0, ...].

%F a(3*n + 1) = 0.

%F Convolution inverse of A262966.

%e G.f. = 1 - 2*q^2 + 2*q^3 - 4*q^5 + 4*q^6 - 6*q^8 + 8*q^9 - 12*q^11 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^2] / EllipticTheta[ 4, 0, q^3], {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A) / (eta(x^3 + A)^2 * eta(x^4 + A)), n))};

%o (PARI) q='q+O('q^99); Vec(eta(q^2)^2*eta(q^6)/(eta(q^3)^2*eta(q^4))) \\ _Altug Alkan_, Jul 31 2018

%Y Cf. A262966.

%K sign

%O 0,3

%A _Michael Somos_, Oct 05 2015