OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (phi(-q^3)^3 / phi(-q)) / (psi(q)^3 / psi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q) / eta(q^6)) * (eta(q^3) / eta(q^2))^5 in powers of q.
Euler transform of period 6 sequence [ -1, 4, -6, 4, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (1-v) * (8-9*u) + (u-v)^2.
G.f.: Product_{k>0} (1 - x^k) / (1 - x^(6*k)) * ((1 - x^(3*k)) / (1 - x^(2*k)))^5.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 2/3 - (2/9)*sqrt(3) + (2/9)*sqrt(6)*3^(1/4). - Simon Plouffe, Mar 04 2021
a(n) = (-1)^n*A164617(n). - Michael Somos, Apr 24 2023
EXAMPLE
G.f. = 1 - q + 4*q^2 - 10*q^3 + 20*q^4 - 39*q^5 + 76*q^6 - 140*q^7 + ...
MATHEMATICA
eta[x_] := x^(1/24)*QPochhammer[x]; A128641[n_] := SeriesCoefficient[ (eta[q]/eta[q^6])*(eta[q^3]/eta[q^2])^5, {q, 0, n}]; Table[A128641[n], {n, 0, 50}] (* G. C. Greubel, Aug 22 2017 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x^2 + A))^5 * eta(x + A) / eta(x^6 + A), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 16 2007
STATUS
approved