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A263527 Expansion of phi(-x^3) * f(-x^6)^3 / f(-x^2) in powers of x where phi(), f() are Ramanujan theta functions.

%I A263527 #10 Feb 16 2025 08:33:27
%S A263527 1,0,1,-2,2,-2,0,-4,2,0,1,-4,4,-2,2,-4,5,0,2,-2,6,-4,2,-4,6,0,0,-6,4,
%T A263527 -2,4,-8,7,0,2,-10,4,-6,0,-4,6,0,1,-6,8,-6,4,-8,4,0,4,-8,10,-4,2,-8,8,
%U A263527 0,2,-6,12,-4,4,-8,8,0,5,-8,6,-4,0,-8,14,0,2,-10
%N A263527 Expansion of phi(-x^3) * f(-x^6)^3 / f(-x^2) in powers of x where phi(), f() are Ramanujan theta functions.
%C A263527 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A263527 G. C. Greubel, <a href="/A263527/b263527.txt">Table of n, a(n) for n = 0..1000</a>
%H A263527 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263527 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263527 Expansion of q^(-2/3) * eta(q^3)^2 * eta(q^6)^2 / eta(q^2) in powers of q.
%F A263527 Euler transform of period 6 sequence [ 0, 1, -2, 1, 0, -3, ...].
%F A263527 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = (2048/3)^(1/2) (t/I)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263501.
%F A263527 a(n) = (-1)^n * A261444(n). a(8*n + 1) = 0.
%F A263527 a(2*n) = A261426(n). a(4*n) = A263433(n). a(4*n + 2) = A261444(n).
%e A263527 G.f. = 1 + x^2 - 2*x^3 + 2*x^4 - 2*x^5 - 4*x^7 + 2*x^8 + x^10 - 4*x^11 + ...
%e A263527 G.f. = q^2 + q^8 - 2*q^11 + 2*q^14 - 2*q^17 - 4*q^23 + 2*q^26 + q^32 + ...
%t A263527 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] QPochhammer[ x^6]^3 / QPochhammer[ x^2], {x, 0, n}];
%o A263527 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^6 + A)^2 / eta(x^2 + A), n))};
%Y A263527 Cf. A261426, A261444, A263433, A263501.
%K A263527 sign
%O A263527 0,4
%A A263527 _Michael Somos_, Oct 19 2015