OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (eta(q) / eta(q^8))^2 * (eta(q^4) / eta(q^2))^7 in powers of q.
Euler transform of period 8 sequence [-2, 5, -2, -2, -2, 5, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A080015.
a(n) ~ (-1)^n * exp(sqrt(n)*Pi) / (8*n^(3/4)). - Vaclav Kotesovec, Nov 17 2017
EXAMPLE
G.f. = 1 - 2*q + 6*q^2 - 12*q^3 + 22*q^4 - 40*q^5 + 68*q^6 - 112*q^7 + 182*q^8 + ...
MATHEMATICA
nmax = 40; CoefficientList[Series[Product[((1 - x^k) / (1 - x^(8*k)))^2 * (1 + x^(2*k))^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 17 2017 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]/ eta[q^8])^2*(eta[q^4]/eta[q^2])^7, {q, 0, 50}], q] (* G. C. Greubel, Aug 11 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^8 + A))^2 * (eta(x^4 + A) / eta(x^2 + A))^7, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 16 2012
STATUS
approved