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Expansion of f(-x, -x) * f(-x^6, -x^6) / f(x, x^2) in powers of x where f(,) is Ramanujan's general theta function.
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%I #11 Mar 12 2021 22:24:48

%S 1,-3,2,1,-1,-1,3,-1,0,-2,-2,2,1,-3,3,4,-1,-3,1,0,-1,-2,0,3,1,-6,2,4,

%T -4,-1,4,2,-1,-3,0,5,-1,-9,5,7,-4,-7,4,5,-3,-4,0,8,-1,-13,4,11,-7,-7,

%U 7,6,-1,-10,0,14,-1,-15,8,15,-10,-14,8,11,-7,-13,2,17

%N Expansion of f(-x, -x) * f(-x^6, -x^6) / f(x, x^2) 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="/A257657/b257657.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 q^(1/24) * eta(q)^3 * eta(q^6)^3 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^12)) in powers of q.

%F Euler transform of period 12 sequence [ -3, -1, -1, -1, -3, -2, -3, -1, -1, -1, -3, -1, ...].

%F a(n) = 2 * A260413(n) - A053250(n).

%e G.f. = 1 - 3*x + 2*x^2 + x^3 - x^4 - x^5 + 3*x^6 - x^7 - 2*x^9 - 2*x^10 + ...

%e G.f. = 1/q - 3*q^23 + 2*q^47 + q^71 - q^95 - q^119 + 3*q^143 - q^167 + ...

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

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

%Y Cf. A053250, A260413.

%K sign

%O 0,2

%A _Michael Somos_, Jul 26 2015