OFFSET
0,8
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 7 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 21 2014
There is a typo in the entry for this q-series in Table I of Yang 2004. The exponent of 18 should be 3. - Michael Somos, Jul 21 2014
A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(18). [Yang 2004] - Michael Somos, Jul 21 2014
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
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
G.f.: Product_{k>0} (1 - x^(6*k - 3))^3 / (1 - x^(2*k-1)). - Michael Somos, Mar 17 2004
Expansion of chi(-q^3)^3 / chi(-q) in powers of q where chi() is a Ramanujan theta function.
Expansion of q^(1/3) * c(q) / c(q^2) in powers of q where c() is a cubic AGM theta function. - Michael Somos, Oct 17 2006
Expansion of q^(1/3) * eta(q^2) * eta(q^3)^3 / (eta(q) * eta(q^6)^3) in powers of q. - Michael Somos, Mar 05 2004
Euler transform of period 6 sequence [ 1, 0, -2, 0, 1, 0, ...]. - Michael Somos, Mar 05 2004
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 2*u +v^2 - u^2*v. - Michael Somos, Mar 17 2004
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 - v + v^2) * u^3 - (4 + 2*v + v^2) * v. - Michael Somos, Aug 11 2007
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^5)) where f(u, v) = u^6 + v^6 - u^5*v^5 + 5*u^4*v^4 - 20*u^3*v^3 + 20*u^2*v^2 - 16*u*v + 5*u^2*v^5 + 5*u^5*v^2 - 10*u^4*v - 10*u*v^4. - Michael Somos, Aug 11 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128128.
G.f.: 1 + x*(1+x)/(1 + x^2*(1+x^2)/(1 + x^3*(1+x^3)/(1 + x^4*(1+x^4)/(1 + x^5*(1+x^5)/(1 + ...))))), a continued fraction. - Paul D. Hanna, Jul 09 2013
Convolution inverse of A092848.
EXAMPLE
G.f. = 1 + x + x^2 - x^3 - x^4 + x^6 + 2*x^7 - 2*x^9 - 3*x^10 - x^11 + 4*x^12 + ...
T18D = 1/q + q^2 + q^5 - q^8 - q^11 + q^17 + 2*q^20 - 2*q^26 - 3*q^29 - x^32 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3]^3 / (QPochhammer[ x] QPochhammer[ x^6]^3), {x, 0, n}]; (* Michael Somos, Jul 21 2014 *)
PROG
(PARI) {a(n) = local(A, m); if( n<0, 0, m=1; A = 1 + O(x); while( m<=n, m*=3; A = subst(A, x, x^3) / x; A = (x*A * (4 + 2*A + A^2) / (1 - A + A^2))^(1/3)); polcoeff(A, n))};
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3), n))};
(PARI) /* Continued Fraction Expansion: */
{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1 + x^(n-k+1)*(1 + x^(n-k+1))/(CF+x*O(x^n))); polcoeff(CF, n)} \\ Paul D. Hanna, Jul 09 2013
CROSSREFS
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Jun 30 2001
STATUS
approved