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A208604 Expansion of phi(-q) / phi(q^4) in powers of q where phi() is a Ramanujan theta function.

%I A208604 #20 Feb 16 2025 08:33:16
%S A208604 1,-2,0,0,0,4,0,0,0,-10,0,0,0,20,0,0,0,-36,0,0,0,64,0,0,0,-110,0,0,0,
%T A208604 180,0,0,0,-288,0,0,0,452,0,0,0,-692,0,0,0,1044,0,0,0,-1554,0,0,0,
%U A208604 2276,0,0,0,-3296,0,0,0,4724,0,0,0,-6696,0,0,0,9408,0,0
%N A208604 Expansion of phi(-q) / phi(q^4) in powers of q where phi() is a Ramanujan theta function.
%C A208604 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A208604 G. C. Greubel, <a href="/A208604/b208604.txt">Table of n, a(n) for n = 0..1000</a>
%H A208604 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208604 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208604 Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^16)^2 / (eta(q^2) * eta(q^8)^5) in powers of q.
%F A208604 Euler transform of period 16 sequence [ -2, -1, -2, -3, -2, -1, -2, 2, -2, -1, -2, -3, -2, -1, -2, 0, ...].
%F A208604 G.f.: (Sum_{k in Z} (-1)^k * x^k^2) / (Sum_{k in Z} x^(4 * k^2)).
%F A208604 a(4*n) = 0 unless n=0. a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = -2 * A079006(n).
%F A208604 a(n) = (-1)^n * A208274(n). Convolution inverse of A208933. - _Michael Somos_, Dec 11 2016
%F A208604 G.f.: Product_{k>0} (1 + x^(8*k)) / ((1 + x^k)^2 * (1 + x^(2*k)) * (1 + x^(4*k))^3). - _Michael Somos_, Dec 11 2016
%e A208604 G.f. = 1 - 2*q + 4*q^5 - 10*q^9 + 20*q^13 - 36*q^17 + 64*q^21 - 110*q^25 + ...
%t A208604 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 3, 0, q^4], {q, 0, n}]; (* _Michael Somos_, Dec 11 2016 *)
%o A208604 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)^2 / (eta(x^2 + A) * eta(x^8 + A)^5), n))};
%Y A208604 Cf. A079006, A208274, A208933.
%K A208604 sign
%O A208604 0,2
%A A208604 _Michael Somos_, Feb 29 2012