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Expansion of phi(-q) / phi(-q^9) in powers of q where phi() is a Ramanujan theta function.
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%I #15 Mar 12 2021 22:24:44

%S 1,-2,0,0,2,0,0,0,0,0,-4,0,0,4,0,0,2,0,0,-8,0,0,8,0,0,2,0,0,-16,0,0,

%T 16,0,0,4,0,0,-28,0,0,28,0,0,8,0,0,-48,0,0,46,0,0,12,0,0,-80,0,0,76,0,

%U 0,20,0,0,-126,0,0,120,0,0,32,0,0,-196,0,0,184,0,0,48,0,0,-300,0,0,280,0,0,72,0,0

%N Expansion of phi(-q) / phi(-q^9) in powers of q where phi() is a Ramanujan theta function.

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

%H G. C. Greubel, <a href="/A128771/b128771.txt">Table of n, a(n) for n = 0..1000</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 eta(q)^2 * eta(q^18) / (eta(q^2) * eta(q^9)^2) in powers of q.

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

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1-u) * (u-v^2) - 2*u * (v-1).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u-v)^3 - u* (3-u) * (v-1) * (3 - 2*u + u*v).

%F G.f.: Product_{k>0} (1 - x^k) * (1 + x^(9*k)) / ( (1 + x^k) * (1 - x^(9*k)) ).

%F a(3n+2)= a(3n+3)= 0.

%F Empirical : sum(exp(-Pi/3)^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = sqrt(3). Simon Plouffe, Feb. 20, 2011.

%F Convolution inverse of A128770. a(3*n + 1) = -2*A092848(n).

%e G.f. = 1 - 2*q + 2*q^4 - 4*q^10 + 4*q^13 + 2*q^16 - 8*q^19 + 8*q^22 + 2*q^25 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^9], {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^18 + A) / (eta(x^2 + A) * eta(x^9 + A)^2), n))};

%Y Cf. A092848, A128770.

%K sign

%O 0,2

%A _Michael Somos_, Mar 27 2007