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 A217771 Expansion of (phi(-x) / phi(-x^3))^2 in powers of x where phi() is a Ramanujan theta function. 4
 1, -4, 4, 4, -12, 8, 12, -32, 20, 28, -72, 48, 60, -152, 96, 120, -300, 184, 228, -560, 344, 416, -1008, 608, 732, -1756, 1048, 1252, -2976, 1768, 2088, -4928, 2900, 3408, -7992, 4672, 5460, -12728, 7408, 8600, -19944, 11544, 13344, -30800, 17744, 20424 (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). 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 eta(q)^4 * eta(q^6)^2 / (eta(q^2)^2 * eta(q^3)^4) in powers of q. Euler transform of period 6 sequence [ -4, -2, 0, -2, -4, 0, ...]. G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u) * (u + v^2) - 4 * u. G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (3 + u * v)^2 - v * (3*u + v)^2. G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A217786. a(n) = - 4 * A123649(n) unless n=0. Convolution inverse of A186924. Convolution square of A139137. EXAMPLE G.f. = 1 - 4*x + 4*x^2 + 4*x^3 - 12*x^4 + 8*x^5 + 12*x^6 - 32*x^7 + 20*x^8 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2 / EllipticTheta[ 4, 0, q^3]^2, {q, 0, n}]; (* Michael Somos, Mar 24 2013 *) PROG (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A)^4), n))} CROSSREFS Cf. A123649, A186924, A217786. Sequence in context: A151896 A267191 A170897 * A261321 A245517 A179526 Adjacent sequences:  A217768 A217769 A217770 * A217772 A217773 A217774 KEYWORD sign AUTHOR Michael Somos, Mar 24 2013 STATUS approved

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Last modified November 28 07:52 EST 2021. Contains 349401 sequences. (Running on oeis4.)