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
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
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Nov 09 2005
STATUS
approved