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

0,10

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

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.

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.

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

Cf. A003823, A083365, A092869.

Sequence in context: A055347 A055288 A203995 * A325181 A072739 A328699

Adjacent sequences: A111371 A111372 A111373 * A111375 A111376 A111377

Keyword

sign

Author

N. J. A. Sloane, Nov 09 2005

Status

approved