OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
FORMULA
Expansion of q^(1/3) * eta(q^3)^4 / (eta(q) * eta(q^9))^2 in powers of q. - Michael Somos, Jun 30 2011
Expansion of q^(1/3) * ( (c(q) * b(q^3)) / (b(q) * c(q^3)) )^(1/2) in powers of q where b(), c() are cubic AGM functions. - Michael Somos, Jun 30 2011
Euler transform of period 9 sequence [ 2, 2, -2, 2, 2, -2, 2, 2, 0, ...]. - Michael Somos, Jun 30 2011
Given g.f. A(x) then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - v) * (u - v^2) + 4*u*v. - Michael Somos, Jun 30 2011
G.f. is a period 1 Fourier series which satisfies f(-1 / (81 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 30 2011
G.f.: Product_{k>0} (1 - x^(3*k))^4 / ((1 - x^k) * (1 - x^(9*k)))^2.
Convolution inverse of A192329.
a(n) ~ exp(4*Pi*sqrt(n/3)/3) / (sqrt(2) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
EXAMPLE
G.f. = 1 + 2*x + 5*x^2 + 6*x^3 + 12*x^4 + 16*x^5 + 27*x^6 + 34*x^7 + ...
T27b = 1/q + 2*q^2 + 5*q^5 + 6*q^8 + 12*q^11 + 16*q^14 + 27*q^17 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3]^2 / (QPochhammer[ x] QPochhammer[ x^9]))^2, {x, 0, n}]; (* Michael Somos, Aug 28 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^4 / (eta(x + A) * eta(x^9 + A))^2, n))}; /* Michael Somos, Jun 30 2011 */
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 27 2000
STATUS
approved