login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A135763 Expansion of (theta_3(q) * theta_3(q^3))^3 in powers of q. 1
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 (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

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: A113791 A292289 A281352 * A030659 A142338 A114304

Adjacent sequences:  A135760 A135761 A135762 * A135764 A135765 A135766

KEYWORD

nonn

AUTHOR

Michael Somos, Nov 28 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 15 13:44 EDT 2021. Contains 343920 sequences. (Running on oeis4.)