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 A080015 Expansion of theta_3(q) / theta_3(q^2) in powers of q. 5
 1, 2, -2, -4, 6, 8, -12, -16, 22, 30, -40, -52, 68, 88, -112, -144, 182, 228, -286, -356, 440, 544, -668, -816, 996, 1210, -1464, -1768, 2128, 2552, -3056, -3648, 4342, 5160, -6116, -7232, 8538, 10056, -11820, -13872, 16248, 18996, -22176, -25844, 30068 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). REFERENCES B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 214 Entry 24(ii). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of phi(q) / phi(q^2) in powers of q where phi() is a Ramanujan theta function. Expansion of eta(q^2)^7 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)^7) in powers of q. Euler transform of period 8 sequence [ 2, -5, 2, 2, 2, -5, 2, 0, ...]. 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 Expansion of phi(x) / phi(x^2) where phi() is a Ramanujan theta function. 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 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 Expansion of sqrt(m) in powers of q where m is the multiplier for the second degree modular equation. 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. a(n) = (-1)^n * A210030(n). a(n) = (-1)^[n/2] * A080054(n). EXAMPLE 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 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Apr 24 2015 *) PROG (PARI) {a(n) = my(A, m); if( n<0, 0, m=1; A = 1 + 2 * x + O(x^2); while( m

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Last modified May 13 00:11 EDT 2021. Contains 343829 sequences. (Running on oeis4.)