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A245643 Expansion of eta(q)^6 * eta(q^2) / eta(q^4)^2 in powers of q.

%I A245643 #66 Feb 16 2025 08:33:23
%S A245643 1,-6,8,16,-38,-16,48,64,-56,-150,112,112,-112,-80,160,192,-294,-288,
%T A245643 248,304,-272,-160,368,320,-336,-726,400,448,-448,-240,544,640,-568,
%U A245643 -864,736,608,-950,-400,656,832,-784,-1152,864,1008,-784,-496,1184,896,-1136
%N A245643 Expansion of eta(q)^6 * eta(q^2) / eta(q^4)^2 in powers of q.
%C A245643 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A245643 G. C. Greubel, <a href="/A245643/b245643.txt">Table of n, a(n) for n = 0..1000</a>
%H A245643 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A245643 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A245643 Expansion of phi(q) * phi(-q)^4 = phi(-q)^3 * phi(-q^2)^2 = phi(-q^2)^8 / phi(q)^3 = f(-q)^6 / psi(q^2) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
%F A245643 Euler transform of period 4 sequence [ -6, -7, -6, -5, ...].
%F A245643 G.f.: Product_{k>0} (1 - x^k)^6 * (1 - x^(2*k)) / (1 - x^(4*k))^2.
%F A245643 Convolution inverse of A134416.
%F A245643 G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 8192^(1/2) (t/I)^(5/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A266575. - _Michael Somos_, Jan 03 2016
%F A245643 a(3*n + 2) = 8 * A263398(n). - _Michael Somos_, Oct 16 2015
%e A245643 G.f. = 1 - 6*q + 8*q^2 + 16*q^3 - 38*q^4 - 16*q^5 + 48*q^6 + 64*q^7 + ...
%t A245643 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q]^4, {q, 0, n}];
%t A245643 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^3 EllipticTheta[ 4, 0, q^2]^2, {q, 0, n}];
%t A245643 a[ n_] := SeriesCoefficient[ 2 q^(1/4) QPochhammer[ q]^6 / EllipticTheta[ 2, 0, q], {q, 0, n}];
%o A245643 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6 * eta(x^2 + A) / eta(x^4 + A)^2, n))};
%o A245643 (Magma) A := Basis( ModularForms( Gamma1(4), 5/2), 49); A[1] - 6*A[2];
%Y A245643 Cf. A134416, A263398, A266575.
%K A245643 sign
%O A245643 0,2
%A A245643 _Michael Somos_, Sep 01 2014