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A266684 Expansion of f(-x) * f(-x^2)^4 / psi(x^3) in powers of x where psi(), f() are Ramanujan theta functions.

%I A266684 #10 Feb 16 2025 08:33:28
%S A266684 1,-1,-5,3,7,4,3,-18,-17,-1,20,36,-9,-14,-18,-12,31,16,-5,-54,-28,6,
%T A266684 36,72,15,-21,-70,3,54,28,-12,-90,-65,-12,80,72,7,-38,-54,42,68,40,30,
%U A266684 -126,-108,4,72,144,-33,-43,-105,-48,98,52,3,-144,-90,18,140,180
%N A266684 Expansion of f(-x) * f(-x^2)^4 / psi(x^3) in powers of x where psi(), f() are Ramanujan theta functions.
%C A266684 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A266684 Alois P. Heinz, <a href="/A266684/b266684.txt">Table of n, a(n) for n = 0..10000</a>
%H A266684 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A266684 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A266684 Expansion of eta(q) * eta(q^2)^4 * eta(q^3) / eta(q^6)^2 in powers of q.
%F A266684 Euler transform of period 6 sequence [ -1, -5, -2, -5, -1, -4, ...].
%F A266684 G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 248832^(1/2) (t/I)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263021.
%F A266684 a(n) = A260301(3*n). a(3*n) = A260301(n).
%e A266684 G.f. = 1 - x - 5*x^2 + 3*x^3 + 7*x^4 + 4*x^5 + 3*x^6 - 18*x^7 - 17*x^8 + ...
%t A266684 a[ n_] := SeriesCoefficient[ 2 q^(3/8) QPochhammer[ q] QPochhammer[ q^2]^4 / EllipticTheta[ 2, 0, q^(3/2)], {q, 0, n}];
%o A266684 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A)^4 * eta(x^3 + A) / eta(x^6 + A)^2, n))};
%Y A266684 Cf. A263021, A260301.
%K A266684 sign,look
%O A266684 0,3
%A A266684 _Michael Somos_, Jan 02 2016