%I #14 Mar 12 2021 22:24:45

%S 1,-1,1,-1,2,-3,4,-5,7,-10,12,-15,20,-26,32,-39,50,-63,76,-92,114,

%T -140,168,-201,244,-295,350,-415,496,-591,696,-818,967,-1140,1332,

%U -1554,1820,-2126,2468,-2861,3324,-3855,4448,-5126,5916,-6816,7824,-8970,10292,-11793,13471,-15372,17548,-20007

%N Expansion of c(q^3) / (c(q^3) + c(q^6)) where c() is a cubic AGM function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. C. Greubel, <a href="/A145977/b145977.txt">Table of n, a(n) for n = 0..500</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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.

%F 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.

%F Euler transform of period 18 sequence [ -1, 1, 0, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, 0, 1, -1, 0, ...].

%F 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.

%F 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.

%F Convolution inverse of A139032.

%F a(n) = - A124243(n) unless n=0. a(2*n) = A128129(n) = a(2*n) unless n=0.

%F a(2*n + 1) = - A132302(n). a(3*n) = A128641(n).

%e 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 + ...

%t a[ n_] := SeriesCoefficient[ 1 - EllipticTheta[ 2, 0, x^(9/2)] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)

%o (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))};

%Y Cf. A124243, A128129, A128641, A132302, A139032.

%K sign

%O 0,5

%A _Michael Somos_, Oct 26 2008