OFFSET
-1,2
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). See Table 4 18B.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (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 ((c(q) * c(q^2) * b(q^3) * b(q^6)) / (b(q) * b(q^2) * c(q^3) * c(q^6)))^(1/2) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of (eta(q^3) * eta(q^6))^4 / (eta(q) * eta(q^2) * eta(q^9) * eta(q^18))^2 in powers of q.
Euler transform of period 18 sequence [ 2, 4, -2, 4, 2, -4, 2, 4, 0, 4, 2, -4, 2, 4, -2, 4, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = f(t) where q = exp(2 Pi i t).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * v * (u + 3) * (v + 3) * (u + v) - (u^2 + 4 * u * v + v^2)^2.
G.f. A(x) = B(x) * B(x^2) where B(x) is g.f. for A058601.
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015
EXAMPLE
1/q + 2 + 7*q + 10*q^2 + 27*q^3 + 38*q^4 + 82*q^5 + 108*q^6 + 207*q^7 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(((1-x^(3*k)) * (1-x^(6*k)))^2 / ((1-x^k) * (1-x^(2*k)) * (1-x^(9*k)) * (1-x^(18*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
QP = QPochhammer; s=(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, adapted from PARI *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^6 + A))^4 / (eta(x + A) * eta(x^2 + A) * eta(x^9 + A) * eta(x^18 + A))^2, n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 09 2012
STATUS
approved