A112148 McKay-Thompson series of class 12B for the Monster group.

1, 0, 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, -17744, -20448, 46944

Offset

-1,3

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, Commun. Algebra 22, No. 13, 5175-5193 (1994).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of -5 + (1/q) * (phi(q)^3 * psi(-q)) / (phi(q^3) * psi(-q^3)^3) in powers of q where phi(), psi() are Ramanujan theta functions.

a(n) = -(-1)^n * A007258(n). - Michael Somos, May 20 2015

a(n) = A187146(n) = A187147(n) = A187148(n) unless n=0. - Michael Somos, May 20 2015

Example

T12B = 1/q + 6*q - 4*q^2 - 3*q^3 + 12*q^4 - 8*q^5 - 12*q^6 + 30*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ -5 + 2 EllipticTheta[3, 0, q]^3 EllipticTheta[2, Pi/4, q^(1/2)] / (EllipticTheta[3, 0, q^3] EllipticTheta[2, Pi/4, q^(3/2)]^3), {q, 0, n}]; (* Michael Somos, May 20 2015 *)
QP = QPochhammer; s = -5*q +QP[q^2]^14/(QP[q]^5*QP[q^3]*QP[q^4]^5* QP[q^6]^2*QP[q^12]) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A =x * O(x^n); polcoeff( -5 * x + eta(x^2 + A)^14 / (eta(x + A)^5 * eta(x^3 + A) * eta(x^4 + A)^5 * eta(x^6 + A)^2 * eta(x^12 + A)), n))}; /* Michael Somos, May 20 2015 */

Crossrefs

Cf. A007258, A187146, A187147, A187148.

Sequence in context: A372365 A020794 A268818 * A007258 A045488 A082530

Adjacent sequences: A112145 A112146 A112147 * A112149 A112150 A112151

Keyword

sign

Author

Michael Somos, Aug 28 2005

Status

approved