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A212885 Expansion of phi(q) * phi(-q)^2 in powers of q where phi() is a Ramanujan theta function.

%I A212885 #18 Feb 16 2025 08:33:17
%S A212885 1,-2,-4,8,6,-8,-8,0,12,-10,-8,24,8,-8,-16,0,6,-16,-12,24,24,-16,-8,0,
%T A212885 24,-10,-24,32,0,-24,-16,0,12,-16,-16,48,30,-8,-24,0,24,-32,-16,24,24,
%U A212885 -24,-16,0,8,-18,-28,48,24,-24,-32,0,48,-16,-8,72,0,-24,-32
%N A212885 Expansion of phi(q) * phi(-q)^2 in powers of q where phi() is a Ramanujan theta function.
%C A212885 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A212885 G. C. Greubel, <a href="/A212885/b212885.txt">Table of n, a(n) for n = 0..1000</a>
%H A212885 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A212885 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A212885 Expansion of phi(-x) * phi(-x^2)^2 = phi(-x^2)^4 / phi(x) in powers of x where phi() is a Ramanujan theta function.
%F A212885 Expansion of eta(q^2)^3 * eta(q)^2 / eta(q^4)^2 in powers of q.
%F A212885 Euler transform of period 4 sequence [-2, -5, -2, -3, ...].
%F A212885 G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A045828.
%F A212885 G.f.: Product_{k>0} (1 - x^(2*k))^3 * (1 - x^k)^2 / (1 - x^(4*k))^2.
%F A212885 a(4*n) = A005875(n). a(4*n + 1) = -2 * A045834(n). a(4*n + 2) = - A005877(n) = -4 * A045828(n).
%F A212885 a(8*n) = A004015(n). a(8*n + 3) = A005878(n) = 8 * A008443(n). a(8*n + 4)= A005887(n). a(8*n + 5) = -2 * A004024(n). a(8*n + 6) = -8 * A213624(n). a(8*n + 7) = 0.
%e A212885 G.f. = 1 - 2*q - 4*q^2 + 8*q^3 + 6*q^4 - 8*q^5 - 8*q^6 + 12*q^8 - 10*q^9 + ...
%t A212885 a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]* EllipticTheta[3, 0, -q]^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Nov 30 2017 *)
%o A212885 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x + A)^2 / eta(x^4 + A)^2, n))};
%Y A212885 Cf. A004015, A004024, A005875, A005877, A005878, A005887, A008443, A045828, A045834, A213624.
%K A212885 sign
%O A212885 0,2
%A A212885 _Michael Somos_, May 29 2012