OFFSET
0,5
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of 1 - q * psi(q^9) / psi(q) = phi(-q^9) / (psi(q) * chi(-q^3)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^6) * eta(q^9)^2 / (eta(q^2)^2 * eta(q^3) * eta(q^18)), in powers of q.
Euler transform of period 18 sequence [ -1, 1, 0, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, 0, 1, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (2/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A139032.
G.f.: Product_{k>0} (P(3, x^k) * P(9, x^k)) / (P(4, x^k)^2 * P(18, x^k)) where P(n, x) is the n-th cyclotomic polynomial.
Convolution inverse of A139032.
EXAMPLE
G.f. = 1 - q + q^2 - q^3 + 2*q^4 - 3*q^5 + 4*q^6 - 5*q^7 + 7*q^8 - 10*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1 - EllipticTheta[ 2, 0, x^(9/2)] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^9 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 26 2008
STATUS
approved