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 A229616 Expansion of (phi(-q)^3 / phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function. 5
 1, -12, 60, -156, 204, -72, -84, -96, 492, -588, 360, -144, 60, -168, 480, -936, 1068, -216, -516, -240, 1224, -1248, 720, -288, 348, -372, 840, -1884, 1632, -360, -504, -384, 2220, -1872, 1080, -576, -372, -456, 1200, -2184, 2952, -504, -672, -528, 2448 (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). Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882). LINKS G. C. Greubel, Table of n, a(n) for n = 0..2500 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018. FORMULA Expansion of (2*a(q^2) - a(q))^2 = b(q)^4 / b(q^2)^2 in powers of q where a(), b() are cubic AGM theta functions. Expansion of (eta(q)^6 * eta(q^6) / (eta(q^2)^3 * eta(q^3)^2))^2 in powers of q. Euler transform of period 6 sequence [-12, -6, -8, -6, -12, -4, ...]. G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 432 (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229615. G.f.: ( Product_{k>0} (1 + x^(3*k)) * (1 - x^k)^3 / ((1 + x^k)^3 * (1 - x^(3*k))))^2. Convolution square of A122859. Conjecture: -3 A122858(n) - A229616(n) + 4 A282031(n) = 0 for all n. - Thomas Baruchel, Jun 23 2018 EXAMPLE G.f. = 1 - 12*q + 60*q^2 - 156*q^3 + 204*q^4 - 72*q^5 - 84*q^6 - 96*q^7 + ... MATHEMATICA a[ n_] := If[n < 1, Boole[ n == 0], -12 Sum[ {1, -7, 10, -7, 1, 2}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]]; a[ n_] := If[n < 1, Boole[ n == 0], -12 Sum[ {1, -3, 4, -3, 1, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]]; a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^6 / EllipticTheta[ 4, 0, q^3]^2, {q, 0, n}]; PROG (PARI) {a(n) = if( n<1, n==0, -12 * sumdiv( n, d, n/d * [2, 1, -7, 10, -7, 1][d%6 + 1]))}; (PARI) {a(n) = if( n<1, n==0, -12 * sumdiv( n, d, d * [0, 1, -3, 4, -3, 1][d%6 + 1]))}; (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^6 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A)^2))^2, n))}; (Sage) A = ModularForms( Gamma0(6), 2, prec=50).basis(); A[0] - 12*A[1] + 60*A[2]; (MAGMA) A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] - 12*A[2] + 60*A[3]; CROSSREFS Cf. A122859, A229615. Sequence in context: A099830 A158443 A153792 * A321465 A000141 A328094 Adjacent sequences:  A229613 A229614 A229615 * A229617 A229618 A229619 KEYWORD sign AUTHOR Michael Somos, Sep 26 2013 STATUS approved

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Last modified October 18 20:28 EDT 2019. Contains 328197 sequences. (Running on oeis4.)