A187147 McKay-Thompson series of class 12B for the Monster group with a(0) = -4.

1, -4, 6, -4, -3, 12, -8, -12, 30, -20, -30, 72, -46, -60, 156, -96, -117, 300, -188, -228, 552, -344, -420, 1008, -603, -732, 1770, -1048, -1245, 2976, -1776, -2088, 4908, -2900, -3420, 7992, -4658, -5460, 12756, -7408, -8583, 19944, -11564, -13344, 30756

Offset

-1,2

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 = -1..1000

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (1/q) * (psi(-q) / psi(-q^3))^4 in powers of q.

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

Euler transform of period 12 sequence [ -4, 0, 0, -4, -4, 0, -4, -4, 0, 0, -4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 9 / f(t) where q = exp(2 Pi i t).

Convolution square of A062243.

G.f.: ( Product_{k>0} (1 - x^(4*k)) * (1 - x^(2*k-1)) / (1 - x^(3*k)) )^4.

Example

G.f. = 1/q - 4 + 6*q - 4*q^2 - 3*q^3 + 12*q^4 - 8*q^5 - 12*q^6 + 30*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(3/2)])^4, {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(eta[q]* eta[q^4]*eta[q^6]/(eta[q^2]*eta[q^3]*eta[q^12]))^4, {q, 0, 60}], q];
Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 06 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^4, n))};

Crossrefs

Cf. A062243, A112148.

Sequence in context: A204817 A199721 A349942 * A128633 A001482 A198493

Adjacent sequences: A187144 A187145 A187146 * A187148 A187149 A187150

Keyword

sign

Author

Michael Somos, Mar 05 2011

Status

approved