%I #17 Mar 12 2021 22:24:44

%S 1,-4,4,0,2,0,-8,0,-1,0,20,0,-2,0,-40,0,3,0,72,0,2,0,-128,0,-4,0,220,

%T 0,-4,0,-360,0,5,0,576,0,8,0,-904,0,-8,0,1384,0,-10,0,-2088,0,11,0,

%U 3108,0,12,0,-4552,0,-15,0,6592,0,-18,0,-9448,0,22,0,13392

%N Expansion of q^(-1) * (phi(-q) / psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

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

%C Number 3 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - _Michael Somos_, Jul 21 2014

%C A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(8). [Yang 2004] - _Michael Somos_, Jul 21 2014

%H G. C. Greubel, <a href="/A131124/b131124.txt">Table of n, a(n) for n = -1..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>

%H Y. Yang, <a href="http://dx.doi.org/10.1112/S0024609304003510">Transformation formulas for generalized Dedekind eta functions</a>, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.

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

%F Euler transform of period 8 sequence [ -4, -2, -4, -4, -4, -2, -4, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A107035.

%F G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = u * (u + 8) * (v + 4) - v^2.

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

%F Convolution inverse of A107035.

%F a(2*n) = 0 unless n=0. a(n) = A029841(n) unless n=0. a(4*n - 1) = A029839(n). a(4*n + 1) = 4 * A079006(n).

%e G.f. = 1/q - 4 + 4*q + 2*q^3 - 8*q^5 - q^7 + 20*q^9 - 2*q^11 - 40*q^13 + 3*q^15 + ...

%t a[ n_] := SeriesCoefficient[ 4 (EllipticTheta[ 4, 0, q] / EllipticTheta[ 2, 0, q^2])^2, {q, 0, n}]; (* _Michael Somos_, Apr 24 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A) / (eta(x^2 + A) * eta(x^8 + A)^2) )^2, n))};

%Y Cf. A029839, A029841, A079006, A107035.

%K sign

%O -1,2

%A _Michael Somos_, Jun 15 2007