OFFSET
-1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1) * (chi(-q^9) / chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.
Expansion of (eta(q^2) * eta(q^9) / (eta(q) * eta(q^18)))^3 in powers of q.
Euler transform of period 18 sequence [ 3, 0, 3, 0, 3, 0, 3, 0, 0, 0, 3, 0, 3, 0, 3, 0, 3, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 - v) * (w^2 - v) - u*w * (6*(1 + v^2) - 10*v).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u*v - u - v)^3 - u*v* (u + v - 1)^3.
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.: (1/x) * (Product_{k>0} P(x^k))^-3 where P(x) = (x^2 - x + 1) * (x^6 - x^3 + 1).
a(n) = A058535(n) unless n = 0.
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
EXAMPLE
G.f. = 1/q + 3 + 6*q + 13*q^2 + 24*q^3 + 42*q^4 + 73*q^5 + 120*q^6 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q, q] / QPochhammer[ -q^9, q^9])^3, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^2] QPochhammer[ q^9] / (QPochhammer[ q] QPochhammer[ q^18]))^3, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax=60; CoefficientList[Series[Product[((1+x^k) / (1+x^(9*k)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^9 + A) / (eta(x + A) * eta(x^18 + A)))^3, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Mar 06 2007
STATUS
approved