%I #12 Jun 20 2018 03:19:14

%S 1,-2,-3,2,0,6,5,0,3,6,0,-18,12,0,21,16,0,6,27,0,-60,34,0,72,51,0,24,

%T 70,0,-168,101,0,183,134,0,54,182,0,-411,240,0,450,322,0,138,416,0,

%U -936,544,0,981,696,0,282,902,0,-1989,1144,0,2070,1462,0,597,1832,0,-4026,2317,0,4098

%N McKay-Thompson series of class 9d for the Monster group with a(0) = -2.

%H G. C. Greubel, <a href="/A152954/b152954.txt">Table of n, a(n) for n = -1..5000</a>

%F Expansion of F(q) - 2 - 3 / F(q) in powers of q where F(q) = (eta(q^9)^2 / (eta(q^3) * eta(q^27)))^2.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v - u^2) * (u - v^2) + 4 * (1 + u + v) * (u + v + u*v).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (81 t)) = f(t) where q = exp(2 Pi i t).

%F a(3*n) = 0 unless n = 0.

%e 1/q - 2 - 3*q + 2*q^2 + 6*q^4 + 5*q^5 + 3*q^7 + 6*q^8 - 18*q^10 + 12*q^11 + ...

%t QP = QPochhammer; F = (QP[q^9]^2/(QP[q^3]*QP[q^27]))^2; s = F - 2*q - 3*(q^2/F) + O[q]^70; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 16 2015, adapted from PARI *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^9 + A)^2 / eta(x^3 + A) / eta(x^27 + A))^2; polcoeff( A - 2 * x - 3 * x^2 / A, n))}

%Y A058096(n) = a(n) unless n = 0. a(3*n - 1) = A058601(n).

%K sign

%O -1,2

%A _Michael Somos_, Dec 15 2008