A187148 McKay-Thompson series of class 12B for the Monster group with a(0) = -3.

1, -3, 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, Commun. Algebra 22, No. 13, (1994), 5175-5193.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (1/q) * (chi(q^3)^3 / chi(q))^3 in powers of q where chi() is a Ramanujan theta function.

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

Convolution cube of A062244. a(3*n) = -3 * A164617(n). a(3*n + 1) = 6 * A132977(n).

G.f.: (Product_{k>0} (1 + x^(6*k-3))^3 / (1 + x^(2*k-1)))^3.

Example

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

Mathematica

a[ n_] := SeriesCoefficient[ (1/q) (QPochhammer[ -q^3, q^6]^3 / QPochhammer[ -q, q^2])^3, {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)

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)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3))^3, n))};

Crossrefs

Cf. A187146, A187147, A062244, A112148, A132977, A164617.

Sequence in context: A197071 A231737 A140072 * A105559 A090038 A308291

Adjacent sequences: A187145 A187146 A187147 * A187149 A187150 A187151

Keyword

sign

Author

Michael Somos, Mar 05 2011

Status

approved