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Expansion of f(x, x)^2 / (f(x^3, x^3) * f(x, x^5)) in powers of x where f(, ) is Ramanujan's general theta function.
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%I #8 Mar 12 2021 22:24:48

%S 1,3,1,-3,-1,0,1,6,0,-6,-3,-3,4,12,1,-12,-6,-3,5,24,1,-24,-10,-6,11,

%T 42,4,-42,-19,-12,17,72,4,-69,-31,-18,31,120,9,-114,-50,-30,46,189,11,

%U -180,-79,-48,77,294,21,-276,-122,-72,112,450,28,-420,-183,-108

%N Expansion of f(x, x)^2 / (f(x^3, x^3) * f(x, x^5)) in powers of x where f(, ) is Ramanujan's general 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="/A263526/b263526.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 f(x)^3 / (f(-x^2) * f(x^3) * f(-x^6)) in powers of x where f() is a Ramanujan theta function.

%F Expansion of q^(1/3) * eta(q^2)^8 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)^4) in powers of q.

%F Euler transform of period 12 sequence [ 3, -5, 2, -2, 3, -2, 3, -2, 2, -5, 3, 0, ...].

%F a(n) = (-1)^n * A132301(n). Convolution inverse of A261325.

%F a(2*n) = A132179(n). a(2*n + 1) = 3 * A092848(n). a(4*n) = A230256(n). a(4*n + 1) = 3 * A233034(n). a(4*n + 2) = A233037(n). a(4*n + 3) = -3 * A216046(n).

%e G.f. = 1 + 3*x + x^2 - 3*x^3 - x^4 + x^6 + 6*x^7 - 6*x^9 - 3*x^10 + ...

%e G.f. = 1/q + 3*q^2 + q^5 - 3*q^8 - q^11 + q^17 + 6*q^20 - 6*q^26 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3 / (QPochhammer[ x^2] QPochhammer[ -x^3] QPochhammer[ x^6]), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^4), n))};

%Y Cf. A092848, A132179, A132301, A216046, A230256, A233034, A233037, A261325

%K sign

%O 0,2

%A _Michael Somos_, Oct 19 2015