A135763 Expansion of (theta_3(q) * theta_3(q^3))^3 in powers of q.

1, 6, 12, 14, 42, 96, 84, 108, 300, 278, 144, 480, 546, 252, 600, 672, 618, 1152, 732, 828, 2016, 1276, 720, 2112, 2100, 1302, 2040, 2078, 2100, 3360, 1872, 1740, 4908, 3360, 1728, 4800, 5082, 2844, 4344, 4684, 3600, 6720, 4200, 3612, 10080, 5856, 3168, 8832

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

M. Koike, Mathieu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (phi(q) * phi(q^3))^3 in powers of q where phi() is a Ramanujan theta function.

Expansion of (eta(q^2) * eta(q^6))^15 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12))^6 in powers of q.

Euler transform of period 12 sequence [ 6, -9, 12, -3, 6, -18, 6, -3, 12, -9, 6, -6, ...].

G.f. is a period-1 Fourier series which satisfies f(-1 / (12 t)) = 1728^(1/2) (t/i)^3 f(t) where q = exp(2 Pi i t).

G.f.: (( Sum_{k in Z} x^(k^2) ) * ( Sum_{k in Z} x^(3*k^2) ))^3.

Example

G.f. = 1 + 6*q + 12*q^2 + 14*q^3 + 42*q^4 + 96*q^5 + 84*q^6 + 108*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3])^3, {q, 0, n}]; (* Michael Somos, Oct 15 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^15 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A))^6, n))};
(Magma) A := Basis( ModularForms( Gamma1(12), 3), 48); A[1] + 6*A[2] + 12*A[3] + 14*A[4] + 42*A[5] + 96*A[6] + 84*A[7] + 108*A[8] + 300*A[9] + 278*A[10] + 144*A[11] + 480*A[12] + 546*A[13]; /* Michael Somos, Oct 15 2015 */

Crossrefs

Sequence in context: A292289 A281352 A351843 * A030659 A142338 A114304

Adjacent sequences: A135760 A135761 A135762 * A135764 A135765 A135766

Keyword

nonn

Author

Michael Somos, Nov 28 2007

Status

approved