A139380 - OEIS

%I #17 Mar 12 2021 22:24:45

%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,

%U 0,0,-20,0,0,-126,0,0,-120,0,0,32,0,0,196,0,0,184

%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).

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

%H G. C. Greubel, <a href="/A139380/b139380.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 1 + 2 * q * chi(q^3) / chi(q^9)^3 in powers of q where chi() is a Ramanujan theta function.

%F Expansion of 1 - 2 * c(q^6) / c(-q^3) in powers of q where c() is a cubic AGM theta function.

%F Expansion of eta(q^2)^5 * eta(q^9)^2 * eta(q^36)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^18)^5) in powers of q.

%F Euler transform of period 36 sequence [ 2, -3, 2, -1, 2, -3, 2, -1, 0, -3, 2, -1, 2, -3, 2, -1, 2, 0, 2, -1, 2, -3, 2, -1, 2, -3, 0, -1, 2, -3, 2, -1, 2, -3, 2, 0, ...].

%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. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261988. - _Michael Somos_, Sep 07 2015

%F G.f.: (1 + 2 * Sum_{k>0} x^k^2) / (1 + 2 * Sum_{k>0} x^(9*k^2)).

%F G.f.: Product_{k>0} (1 - x^(2*k)) * (1 + x^(2*k-1))^2 / ((1 - x^(18*k)) * (1 + x^(18*k-9))^2).

%F a(n) = A128771(n) unless n=0. a(n) = (-1)^n * A128771(n).

%F a(3*n) = 0 unless n=0. a(3*n + 2) = 0. a(3*n + 1) = 2 * A128111(n).

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

%F Convolution inverse is A261988. - _Michael Somos_, Sep 07 2015

%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[ 3, 0, q] / EllipticTheta[ 3, 0, q^9], {q, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^9 + A)^2 * eta(x^36 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^18 + A)^5), n))};

%Y Cf. A128111, A128771, A261988.

%K sign

%O 0,2

%A _Michael Somos_, Apr 15 2008