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Expansion of q^(-1) * (f(-q^3, -q^5) / f(-q, -q^7))^2 in powers of x where f(,) is Ramanujan's two-variable theta function.
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%I #29 Oct 17 2023 09:18:43

%S 1,2,3,2,1,-2,-4,-4,0,6,9,8,-1,-12,-20,-16,1,22,38,30,1,-40,-64,-52,

%T -2,68,107,88,-2,-112,-180,-144,3,182,292,228,4,-286,-452,-356,-4,440,

%U 686,544,-5,-668,-1044,-816,5,996,1563,1210,6,-1464,-2276,-1768,-8

%N Expansion of q^(-1) * (f(-q^3, -q^5) / f(-q, -q^7))^2 in powers of x where f(,) is Ramanujan's two-variable theta function.

%C Number 10 of the 15 generalized eta-quotients listed in Table I of Yang 2004.

%C A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(8). [Yang, 2004]

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

%H G. A. Edgar, <a href="/A245436/b245436.txt">Table of n, a(n) for n = -1..1003</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 q^(-1) * (psi(-q) * psi(q^4) / f(-q, -q^7)^2)^2 in powers of q where psi() is a Ramanujan theta function.

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

%F Given g.f. A(x), then 0 = f(A(q), A(q^2)) where f(u, v) = (v - 1) * (u^2 - v) - 4*u*v.

%F Convolution inverse of A230535.

%F a(2*n) = 2 * A224216(n+1) unless n=0.

%F a(2*n) = 2 * A245432(n).

%F a(2*n - 1) = A245434(n).

%F a(4*n - 1) = A245433(n).

%F a(4*n + 2) = 2 * A210063(n).

%F G.f.: T(q) = 1/q + 2 + 3*q + ... for this sequence is quadratically related to T8E(q) of A029841: T8E = T + 2 + 1/T. - _G. A. Edgar_, Apr 12 2017

%e G.f. = 1/q + 2 + 3*q + 2*q^2 + q^3 - 2*q^4 - 4*q^5 - 4*q^6 + 6*q^8 + ...

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^3, q^8] QPochhammer[ q^5, q^8] / (QPochhammer[ q^1, q^8] QPochhammer[ q^7, q^8]))^2, {q, 0, n}];

%o (PARI) {a(n) = if( n<-1, 0, n++; polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -2, 0, 2, 0, 2, 0, -2][k%8 + 1]), n))};

%o (PARI) {a(n) = local(A, A2); if( n<-1, 0, n++; A = x * O(x^n); A = eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3; A2 = subst(A, x, x^2); polcoeff( ((A^2 + A2) / (2 * A^2 * A2^2))^2, n))}

%Y Cf. A112143, A224216, A230535, A245432, A245433, A245434.

%K sign

%O -1,2

%A _Michael Somos_, Jul 21 2014