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A263501 Expansion of phi(-x) * f(-x^2)^3 / f(-x^3) in powers of x where phi(), f() are Ramanujan theta functions.

%I A263501 #15 Feb 16 2025 08:33:27
%S A263501 1,-2,-3,7,0,-3,7,-12,-6,12,-2,-3,12,0,-9,13,-12,-9,12,-12,-6,13,0,-6,
%T A263501 24,-12,-6,24,-14,-15,12,0,-9,12,-24,-9,19,0,-12,24,0,-12,36,-24,-9,
%U A263501 19,-12,-12,24,0,-9,12,-36,-15,24,-14,-9,36,0,-18,24,-12,-18
%N A263501 Expansion of phi(-x) * f(-x^2)^3 / f(-x^3) in powers of x where phi(), f() are Ramanujan theta functions.
%C A263501 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A263501 G. C. Greubel, <a href="/A263501/b263501.txt">Table of n, a(n) for n = 0..2500</a>
%H A263501 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263501 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263501 Expansion of q^(-1/8) * eta(q)^2 * eta(q^2)^2 / eta(q^3) in powers of q.
%F A263501 Euler transform of period 6 sequence [ -2, -4, -1, -4, -2, -3, ...].
%F A263501 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 4374^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263527.
%F A263501 -2 * a(n) = A263456(8*n + 1). a(3*n + 2) = -3 * A212907(n). a(9*n + 4) = 0.
%e A263501 G.f. = 1 - 2*x - 3*x^2 + 7*x^3 - 3*x^5 + 7*x^6 - 12*x^7 - 6*x^8 + ...
%e A263501 G.f. = q - 2*q^9 - 3*q^17 + 7*q^25 - 3*q^41 + 7*q^49 - 12*q^57 + ...
%t A263501 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x^2]^3 / QPochhammer[ x^3], {x, 0, n}];
%o A263501 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A)^2 / eta(x^3 + A), n))};
%Y A263501 Cf. A212907, A263456, A263527.
%K A263501 sign
%O A263501 0,2
%A A263501 _Michael Somos_, Oct 19 2015