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%I #59 Sep 08 2022 08:45:13
%S 1,-1,0,1,-1,1,0,-2,2,-1,0,2,-3,2,0,-2,4,-4,0,4,-6,5,0,-6,9,-6,0,7,
%T -12,9,0,-10,16,-13,0,15,-22,17,0,-20,29,-21,0,25,-38,28,0,-32,50,-39,
%U 0,43,-64,49,0,-56,82,-60,0,69,-105,78,0,-86,132,-101,0,112,-166,125,0,-142,208,-153,0,172,-258,192,0
%N Series expansion of the Ramanujan-Goellnitz-Gordon continued fraction.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Glaisher (1876) writes "XIII. tan(pi/16) = (e^(-pi/2) - e^(-3 pi/2) - e^(-15 pi/2) + e^(-21 pi/2) + e^(-45 pi/2) - &c.) / (1 - e^(-6 pi/2) - e^(-10 pi/2) + e^(-28 pi/2) + e^(-36 pi/2) - &c.), ..." where the numerator is q * f( -q^2, -q^14) and denominator is f( -q^6, -q^10) where q = e^(-pi/2). - _Michael Somos_, Jun 22 2012
%C Berndt writes "[...] v = q^(1/2) f(-q,-q^7) / f(-q^3,-q^5). Then v = q^(1/2) / (1 + q + q^2 / (1 + q^3 + q^4 / (1 + q^5 + q^6 / (1 + x^7 + ...)))). (1.1)". - _Michael Somos_, Jul 09 2012
%C Jacobi writes "(7.) (1 - sqrt(k')) / (1 + sqrt(k') + sqrt(2(1+k'))) = (q - q^3 - q^15 + q^21 + q^45 - q^55 - ...) / (1 - q^6 - q^10 + q^28 + q^36 - q^66 - ...)." - _Michael Somos_, Sep 11 2012
%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag; see p. 221 Entry 1(ii), eq. (1.1).
%D J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), 111-112. see Eq. XIII
%D C. G. J. Jacobi, Über die Zur Numerischen Berechnung der Elliptischen Functionen Zweckmaessigsten Formeln, Crelle Bd. 26 (1843), 93-114 = Gesammelte Werke, Bd. 1, 1881, 343-368.
%H G. C. Greubel, <a href="/A092869/b092869.txt">Table of n, a(n) for n = 0..1000</a>
%H S.-D. Chen and S.-S. Huang, <a href="https://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="https://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 W. Duke, <a href="https://dx.doi.org/10.1090/S0273-0979-05-01047-5">Continued fractions and modular functions</a>, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eqs. (9.2), (9.4).
%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 f(-x, -x^7) / f(-x^3, -x^5) in powers of x where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Aug 02 2011
%F Expansion of (phi(x) - phi(x^2)) / (2 * x * psi(x^4)) = 2 * 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 q^(-1) * (1 - sqrt(k')) / (1 + sqrt(k') + sqrt(2 * (1 + k'))) in powers of q^2 where k' is the complementary elliptic modulus. - _Michael Somos_, Sep 11 2012
%F Euler transform of period 8 sequence [-1, 0, 1, 0, 1, 0, -1, 0, ...].
%F G.f. A(x) satisfies both A(-x) * A(x) = A(x^2) and x * A(x)^2 = B(x * A(x^2)) where B(x) = x * (1 - x) / (1 + x).
%F Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v + v^2 + v*u^2.
%F Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (1 - u*v) * (u + v)^3 - v * (1 + v^2) * (1 - u^4). - _Michael Somos_, Feb 15 2006
%F Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^5)) where f(u, v) = (u - v) * (1 + u*v)^5 - u * (1 - u^4) * (1 + v^2) * (1 - 6*v^2 + v^4). - _Michael Somos_, Feb 15 2006
%F G.f.: Product_{k>=0} (1 - x^(8*k + 1)) * (1 - x^(8*k + 7)) / ((1 - x^(8*k + 3)) * (1 - x^(8*k + 5))).
%F G.f. = continued fraction 1/(1 + x + x^2/(1 + x^3 + x^4/(1 + x^5 + x^6/(1 + x^7 + ...)))). Convolution inverse of A111374.
%F a(2*n + 1) = -A226559(n). - _Michael Somos_, Jun 12 2013
%F a(4*n) = A083365(n). a(4*n + 2) = 0.
%F G.f. A(x) satisfies x*A(-x^2) = x*B(x^2)/C(x^2) = (F(x) - F(-x))/(F(x) + F(-x)), where B(x) is the g.f. of A069911, C(x) is the g.f. of A069910 and F(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700. - _Peter Bala_, Feb 07 2021
%e G.f. = 1 - x + x^3 - x^4 + x^5 - 2*x^7 + 2*x^8 - x^9 + 2*x^11 - 3*x^12 + ...
%e G/f. = q - q^3 + q^7 - q^9 + q^11 - 2*q^15 + 2*q^17 - q^19 + 2*q^23 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^8] QPochhammer[ x^7, x^8] /(QPochhammer[ x^3, x^8] QPochhammer[ x^5, x^8]), {x, 0, n}] (* _Michael Somos_, Aug 02 2011 *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^2] / (EllipticTheta[ 3, 0, x] + EllipticTheta[ 3, 0, x^2]), {x, 0, n + 1/2}] (* _Michael Somos_, Aug 02 2011 *)
%t a[ n_] := SeriesCoefficient[ Product[(1 - q^k)^KroneckerSymbol[ 8, k], {k, n}], {q, 0, n}] (* _Michael Somos_, Jul 08 2012 *)
%o (PARI) {a(n) = local(A, u, v); if( n<0, 0, n = 2*n + 1; A = x; forstep( k=3, n, 2, u = A + x * O(x^k); v = subst(u, x, x^2); A -= x^k * polcoeff(u^2 - v + v*u^2 + v^2, k+1) / 2); polcoeff(A, n))}
%o (PARI) {a(n) = local(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = x * subst(A, x, x^2); A = sqrt(A * (1 - A) / (1 + A) / x)); polcoeff(A, n))}
%o (PARI) {a(n) = local(A, A2); if( n<0, 0, A = eta(x^8 + x * O(x^n))^2 / eta(x^4 + x * O(x^n)); A2 = sum( k=1, sqrtint(n), x^k^2 + x^(2*k^2), 1 + x * O(x^n)); polcoeff(A / A2, n))}
%o (PARI) {a(n) = local(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( 2 * A^2 * A2^2 / (A^2 + A2), n))}
%o (PARI) N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^kronecker(2, k))) \\ _Seiichi Manyama_, Sep 24 2019
%o (Magma) A := Basis( ModularForms( Gamma1(16), 1/2), 159); LS<q> := LaurentSeriesRing( RationalField()); A[3] / A[2]; /* _Michael Somos_, Aug 31 2018 */
%Y Cf. A000700, A003823, A083365, A091188, A111374, A226559.
%K sign
%O 0,8
%A _Michael Somos_, Mar 07 2004; corrected Jun 09 2004
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