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A262938 Expansion of phi(-q) / phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

%I A262938 #15 Feb 16 2025 08:33:27
%S A262938 1,-2,0,0,2,0,2,-4,0,-2,4,0,4,-8,0,-4,10,0,8,-16,0,-8,20,0,14,-30,0,
%T A262938 -16,36,0,24,-52,0,-28,64,0,42,-88,0,-48,108,0,68,-144,0,-80,176,0,
%U A262938 108,-230,0,-128,280,0,170,-360,0,-200,436,0,260,-552,0,-308,666
%N A262938 Expansion of phi(-q) / phi(-q^6) in powers of q where phi() is a Ramanujan theta function.
%C A262938 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262938 G. C. Greubel, <a href="/A262938/b262938.txt">Table of n, a(n) for n = 0..1000</a>
%H A262938 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262938 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262938 Expansion of eta(q)^2 * eta(q^12) / (eta(q^2) * eta(q^6)^2) in powers of q.
%F A262938 Euler transform of period 12 sequence [ -2, -1, -2, -1, -2, 1, -2, -1, -2, -1, -2, 0, ...].
%F A262938 G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 6^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A262160.
%F A262938 a(n) = (-1)^n * A143068(n). a(2*n) = A260256(n). a(2*n + 1) = -2 * A261877(n).
%F A262938 a(3*n) = 0.
%e A262938 G.f. = 1 - 2*q + 2*q^4 + 2*q^6 - 4*q^7 - 2*q^9 + 4*q^10 + 4*q^12 + ...
%t A262938 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^6], {q, 0, n}];
%o A262938 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^12 + A) / (eta(x^2 + A) * eta(x^6 + A)^2), n))};
%Y A262938 Cf. A143068, A260256, A261877, A262160.
%K A262938 sign
%O A262938 0,2
%A A262938 _Michael Somos_, Oct 04 2015