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 (psi(q)^3 / psi(q^3)) / (phi(-q^3)^3 / phi(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q^6) / eta(q)) * (eta(q^2) / eta(q^3))^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) = v* (1-v)* (9-8*u) + (u-v)^2.
G.f.: Product_{k>0} (1 - x^(6*k)) / (1 - x^k) * ((1 - x^(2*k)) / (1 - x^(3*k)))^5.
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -(3/8)*sqrt(3) + (3/8)*sqrt(9 + 6*sqrt(3)). - Simon Plouffe, Mar 02 2021
EXAMPLE
G.f. = 1 + q - 3*q^2 + 3*q^3 + 5*q^4 - 18*q^5 + 15*q^6 + 24*q^7 - 75*q^8 + ...
MATHEMATICA
eta[x_] := x^(1/24)*QPochhammer[x]; A128636[n_] := SeriesCoefficient[(eta[q^6]/eta[q])*(eta[q^2]/eta[q^3])^5, {q, 0, n}]; Table[A128636[n], {n, 0, 50}] (* G. C. Greubel, Aug 21 2017 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x^3 + A))^5 * eta(x^6 + A) / eta(x + A), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 16 2007
STATUS
approved