OFFSET
-1,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
From Michael Somos, Aug 19 2012: (Start)
Expansion of -2 + (1/q) * psi(q^3)^2 / (psi(q) * phi(q^9)) * (f(-q^3)^2 / (f(-q) * f(-q^9)))^3 in powers of q where psi(), f() are Ramanujan theta functions.
Expansion of -3 + (1/q) * (chi(-q^9) / chi(-q))^3 + q * (chi(-q) / chi(-q^9))^3 in powers of q where chi() is a Ramanujan theta function.
Expansion of -2 + (eta(q^3) * eta(q^6))^4 / (eta(q) * eta(q^2) * eta(q^9) * eta(q^18))^2 in powers of q.
Expansion of -5 + (eta(q^3)^8 + 4 * eta(q^6)^8) /(eta(q) * eta(q^2) * eta(q^3)^2 * eta(q^6)^2 * eta(q^9) * eta(q^18)).
a(n) = A215407(n) unless n=0. (End)
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015
EXAMPLE
T18B = 1/q + 7*q + 10*q^2 + 27*q^3 + 38*q^4 + 82*q^5 + 108*q^6 + 207*q^7 + ...
MATHEMATICA
QP = QPochhammer; s = -2*q+(QP[q^3]*QP[q^6])^4/(QP[q]*QP[q^2]*QP[q^9]* QP[q^18])^2 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 13 2015, after 3rd formula *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = eta(x + A) * eta(x^2 + A) / (eta(x^9 + A) * eta(x^18 + A)); polcoeff( x + A + 9 * x^2 / A, n))} /* Michael Somos, Aug 19 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 27 2000
STATUS
approved