A229616 Expansion of (phi(-q)^3 / phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function.

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

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

Michael Somos, Introduction to Ramanujan theta functions

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: A361568 A158443 A153792 * A321465 A000141 A328094

Adjacent sequences: A229613 A229614 A229615 * A229617 A229618 A229619

Keyword

sign

Author

Michael Somos, Sep 26 2013

Status

approved