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A260301 Expansion of f(-x^3)^3 * psi(x)^3 / psi(x^3)^2 in powers of x where phi(), f() are Ramanujan theta functions.

%I A260301 #15 Feb 16 2025 08:33:26
%S A260301 1,3,3,-1,-9,-12,-5,6,15,3,-12,-12,7,42,30,4,-33,-48,3,18,36,-18,-60,
%T A260301 -24,-17,63,42,-1,-42,-84,20,30,63,36,-48,-24,-9,114,90,-14,-60,-120,
%U A260301 -18,42,84,-12,-120,-48,31,129,63,16,-126,-156,-5,48,102,-54,-84
%N A260301 Expansion of f(-x^3)^3 * psi(x)^3 / psi(x^3)^2 in powers of x where phi(), f() are Ramanujan theta functions.
%C A260301 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260301 G. C. Greubel, <a href="/A260301/b260301.txt">Table of n, a(n) for n = 0..1000</a>
%H A260301 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260301 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260301 Expansion of eta(q^2)^6 * eta(q^3)^5 / (eta(q)^3 * eta(q^6)^4) in powers of q.
%F A260301 Euler transform of period 6 sequence [ 3, -3, -2, -3, 3, -4, ...].
%F A260301 G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 27648^(1/2) (t/I)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261445.
%e A260301 G.f. = 1 + 3*x + 3*x^2 - x^3 - 9*x^4 - 12*x^5 - 5*x^6 + 6*x^7 + 15*x^8 + ...
%t A260301 a[ n_] := SeriesCoefficient[ (1/2) x^(3/8) QPochhammer[ x^3]^3 EllipticTheta[ 2, 0, x^(1/2)]^3 / EllipticTheta[ 2, 0, x^(3/2)]^2, {x, 0, n}];
%o A260301 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^3 + A)^5 / (eta(x + A)^3 * eta(x^6 + A)^4), n))};
%Y A260301 Cf. A261445.
%K A260301 sign
%O A260301 0,2
%A A260301 _Michael Somos_, Nov 10 2015