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Expansion of phi(x^3) * psi(x^3) / f(x) in powers of x where phi(), psi(), f() are Ramanujan theta functions.
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%I #12 Mar 12 2021 22:24:48

%S 1,-1,2,0,2,-1,4,-2,5,-2,6,-2,10,-4,12,-4,15,-6,20,-8,26,-9,32,-12,40,

%T -16,50,-18,60,-22,76,-28,92,-33,110,-40,134,-50,160,-58,191,-70,230,

%U -84,272,-98,320,-116,380,-138,446,-160,522,-188,612,-222,715,-256

%N Expansion of phi(x^3) * psi(x^3) / f(x) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A260574/b260574.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 f(x^3)^3 / (f(x) * phi(-x^6)) in powers of x where phi(), f() are Ramanujan theta functions.

%F Expansion of q^(-1/3) * eta(q) * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^3 * eta(q^12)^2) in powers of q.

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

%F a(2*n) = A085140(n). a(2*n + 1) = - A097196(n). a(4*n + 1) = - A257655(n).

%e G.f. = 1 - x + 2*x^2 + 2*x^4 - x^5 + 4*x^6 - 2*x^7 + 5*x^8 - 2*x^9 + ...

%e G.f. = q - q^4 + 2*q^7 + 2*q^13 - q^16 + 4*q^19 - 2*q^22 + 5*q^25 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3]^3 / (QPochhammer[ -x] EllipticTheta[ 4, 0, x^6]), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^3 * eta(x^12 + A)^2), n))};

%o (PARI) q='q+O('q^99); Vec(eta(q)*eta(q^4)*eta(q^6)^7/(eta(q^2)^3*eta(q^3)^3*eta(q^12)^2)) \\ _Altug Alkan_, Aug 01 2018

%Y Cf. A085140, A097196, A257655.

%K sign

%O 0,3

%A _Michael Somos_, Jul 29 2015