A080015 - OEIS

%I #18 Mar 12 2021 22:24:42

%S 1,2,-2,-4,6,8,-12,-16,22,30,-40,-52,68,88,-112,-144,182,228,-286,

%T -356,440,544,-668,-816,996,1210,-1464,-1768,2128,2552,-3056,-3648,

%U 4342,5160,-6116,-7232,8538,10056,-11820,-13872,16248,18996,-22176,-25844,30068

%N Expansion of theta_3(q) / theta_3(q^2) in powers of q.

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

%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 214 Entry 24(ii).

%H G. C. Greubel, <a href="/A080015/b080015.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 phi(q) / phi(q^2) in powers of q where phi() is a Ramanujan theta function.

%F Expansion of eta(q^2)^7 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)^7) in powers of q.

%F Euler transform of period 8 sequence [ 2, -5, 2, 2, 2, -5, 2, 0, ...].

%F G.f.: A(x)/B(x), where A(x) = Sum_{m = -infinity..infinity} x^(m^2) and B(x) = Sum_{m = -infinity..infinity} x^(2*m^2). - _Vladeta Jovovic_, Mar 22 2005

%F Expansion of phi(x) / phi(x^2) where phi() is a Ramanujan theta function.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 1 + (1 - u*v)^2 - v^2. - _Michael Somos_, Jan 31 2006

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^4 - v^4 + 8 * u*v - 6 * u*v * (u^2 + v^2) + 4 * (u*v)^3. - _Michael Somos_, Jan 31 2006

%F Expansion of sqrt(m) in powers of q where m is the multiplier for the second degree modular equation.

%F G.f.: Prod_{k>0} ((1 - x^(8*k - 2)) * (1 - x^(8*k - 6)))^5 / ((1 - x^(8*k - 1)) * (1 - x^(8*k - 3)) * (1 - x^(8*k - 4)) * (1 - x^(8*k - 5)) * (1 - x^(8*k - 7)))^2.

%F a(n) = (-1)^n * A210030(n). a(n) = (-1)^[n/2] * A080054(n).

%e G.f. = 1 + 2*q - 2*q^2 - 4*q^3 + 6*q^4 + 8*q^5 - 12*q^6 - 16*q^7 + 22*q^8 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* _Michael Somos_, Apr 24 2015 *)

%o (PARI) {a(n) = my(A, m); if( n<0, 0, m=1; A = 1 + 2 * x + O(x^2); while( m<n, m*=2; A = subst(A, x, x^2); A = (1 + 2 * sqrt((A^2 - 1) / 4)) / A); polcoeff( A, n))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) / eta(x + A))^2 * (eta(x^2 + A) / eta(x^4 + A))^7, n))};

%Y Cf. A080054, A210030.

%K sign,easy

%O 0,2

%A _Michael Somos_, Jan 20 2003