%I #10 Sep 08 2022 08:46:13

%S 1,-4,0,16,-16,8,0,-96,112,44,0,176,-448,-88,0,-32,1136,-200,0,-176,

%T -2016,384,0,224,3136,484,0,-608,-5504,-792,0,640,9328,-704,0,192,

%U -12112,648,0,352,14112,792,0,-208,-21312,-88,0,-2112,31808,-932,0,800

%N Expansion of (eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2)^2 in powers of q.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A259491/b259491.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

%F Euler transform of period 8 sequence [ -4, -6, -4, -12, -4, -6, -4, -8, ...].

%F G.f.: Product_{k>0} ((1 - x^k)^4 * (1 + x^k)^2 * (1 + x^(2*k)) / (1 + x^(4*k))^2)^2.

%F a(2*n + 1) = -4 * A030211(n). a(4*n) = A035016(n). a(4*n + 2) = 0.

%F Convolution square of A131999.

%e G.f. = 1 - 4*q + 16*q^3 - 16*q^4 + 8*q^5 - 96*q^7 + 112*q^8 + 44*q^9 + ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^3 / QPochhammer[ q^8]^2)^2, {q, 0, n}];

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q]^2)^2, {q, 0, n}];

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^4]^2)^2, {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^2 + A) * eta(x^4 + A)^3 / eta(x^8 + A)^2)^2, n))};

%o (Magma) A := Basis( ModularForms( Gamma1(8), 4), 52); A[1] - 4*A[2] + 16*A[4] - 16*A[5] + 8*A[6] - 96*A[8] + 112*A[9];

%Y Cf. A030211, A035016, A131999.

%K sign

%O 0,2

%A _Michael Somos_, Jun 28 2015