A111374 Series expansion of the reciprocal of the Goellnitz-Gordon continued fraction.

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, -12, -9, 0, 12, 16, 13, 0, -14, -22, -17, 0, 18, 29, 21, 0, -26, -38, -28, 0, 34, 50, 39, 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

Offset: 0

Author

N. J. A. Sloane, Nov 09 2005

Keywords

• sign

Comments

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

Number 15 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Aug 07 2014

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

Examples

			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 + ...
G.f. = 1/q + q + q^3 - q^9 - q^11 + q^15 + 2*q^17 + q^19 - 2*q^23 - 3*q^25 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• S.-D. Chen and S.-S. Huang, On the series expansion of the Göllnitz-Gordon continued fraction, Internat. J. Number Theory, 1 (2005), 53-63.
• B. Cho, J. K. Koo, and Y. K. Park, Arithmetic of the Ramanujan-Göllnitz-Gordon continued fraction, J. Number Theory, 129 (2009), 922-947.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
• Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.

Crossrefs

Cf. A003823, A083365, A092869.

Programs

Maple

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(%);

Mathematica

a[ n_] := SeriesCoefficient[ Product[(1 - x^k)^-KroneckerSymbol[ 2, k], {k, n}], {x, 0, n}]; (* Michael Somos, Jul 08 2012 *)
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 *)
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 *)

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 */

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 */

Formula

Expansion of 1 + x + x^2/(1 + x^3 + x^4/(1 + x^5 + x^6/(1 + x^7+ ...))) in powers of x.

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)).

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

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

Euler transform of period 8 sequence [ 1, 0, -1, 0, -1, 0, 1, 0, ...]. - Michael Somos, Mar 08 2012

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

a(4*n + 3) = 0. a(4*n + 1) = A083365(n). Convolution inverse of A092869.