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%I #33 Mar 12 2021 22:24:43
%S 1,1,1,0,0,-1,-1,0,1,2,1,0,-2,-3,-2,0,3,4,4,0,-4,-6,-5,0,5,9,6,0,-8,
%T -12,-9,0,12,16,13,0,-14,-22,-17,0,18,29,21,0,-26,-38,-28,0,34,50,39,
%U 0,-42,-64,-49,0,53,82,60,0,-70,-105,-78,0,90,132,101,0,-110,-166,-125,0,137,208,153,0,-174,-258,-192,0,217
%N Series expansion of the reciprocal of the Goellnitz-Gordon continued fraction.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Number 15 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - _Michael Somos_, Aug 07 2014
%C A generator (Hauptmodul) of the function field associated with the intersection of congruence subgroups Gamma(2) and Gamma_1(8). [Yang 2004] - _Michael Somos_, Aug 07 2014
%H G. C. Greubel, <a href="/A111374/b111374.txt">Table of n, a(n) for n = 0..1000</a>
%H S.-D. Chen and S.-S. Huang, <a href="http://dx.doi.org/10.1142/S1793042105000030">On the series expansion of the Göllnitz-Gordon continued fraction</a>, Internat. J. Number Theory, 1 (2005), 53-63.
%H B. Cho, J. K. Koo, and Y. K. Park, <a href="http://dx.doi.org/10.1016/j.jnt.2008.09.018">Arithmetic of the Ramanujan-Göllnitz-Gordon continued fraction</a>, J. Number Theory, 129 (2009), 922-947.
%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 1 + x + x^2/(1 + x^3 + x^4/(1 + x^5 + x^6/(1 + x^7+ ...))) in powers of x.
%F Let qf(a, q) = Product(1-a*q^j, j=0..infinity); g.f. is qf(q^3, q^8)*qf(q^5, q^8)/(qf(q, q^8)*qf(q^7, q^8)).
%F Expansion of (phi(x) + phi(x^2)) / (2 * psi(x^4)) = 2 * x * psi(x^4) / (phi(x) - phi(x^2)) in powers of x where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Feb 15 2006
%F Expansion of f(-x^3, -x^5) / f(-x, -x^7) in powers of x where f(,) is Ramanujan's two-variable theta function. - _Michael Somos_, Mar 08 2012
%F Euler transform of period 8 sequence [ 1, 0, -1, 0, -1, 0, 1, 0, ...]. - _Michael Somos_, Mar 08 2012
%F Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 * (v - 1) - v * (v + 1). - _Michael Somos_, Oct 22 2013
%F a(4*n + 3) = 0. a(4*n + 1) = A083365(n). Convolution inverse of A092869.
%e G.f. = 1 + x + x^2 - x^5 - x^6 + x^8 + 2*x^9 + x^10 - 2*x^12 - 3*x^13 - 2*x^14 + ...
%e G.f. = 1/q + q + q^3 - q^9 - q^11 + q^15 + 2*q^17 + q^19 - 2*q^23 - 3*q^25 + ...
%p M:=100; qf:=(a,q)->mul(1-a*q^j,j=0..M); t2:=qf(q^3,q^8)*qf(q^5,q^8)/(qf(q,q^8)*qf(q^7,q^8)); series(%,q,M); seriestolist(%);
%t a[ n_] := SeriesCoefficient[ Product[(1 - x^k)^-KroneckerSymbol[ 2, k], {k, n}], {x, 0, n}]; (* _Michael Somos_, Jul 08 2012 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^8] QPochhammer[ x^5, x^8] / (QPochhammer[ x, x^8] QPochhammer[ x^7, x^8] ), {x, 0, n}]; (* _Michael Somos_, Jul 08 2012 *)
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x] + EllipticTheta[ 3, 0, x^2]) / EllipticTheta[ 2, 0, x^2], {x, 0, n - 1/2}]; (* _Michael Somos_, Jul 08 2012 *)
%o (PARI) {a(n) = my(A, A2); if( n<0, 0, 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 ), n))}; /* _Michael Somos_, Mar 08 2012 */
%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n))^-kronecker( 2, k)), n))}; /* _Michael Somos_, Jul 08 2012 */
%Y Cf. A003823, A083365, A092869.
%K sign
%O 0,10
%A _N. J. A. Sloane_, Nov 09 2005
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