A152954 McKay-Thompson series of class 9d for the Monster group with a(0) = -2.

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, 70, 0, -168, 101, 0, 183, 134, 0, 54, 182, 0, -411, 240, 0, 450, 322, 0, 138, 416, 0, -936, 544, 0, 981, 696, 0, 282, 902, 0, -1989, 1144, 0, 2070, 1462, 0, 597, 1832, 0, -4026, 2317, 0, 4098

Offset

-1,2

Links

G. C. Greubel, Table of n, a(n) for n = -1..5000

Formula

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.

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).

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

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

Example

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 + ...

Mathematica

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 *)

Prog

(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))}

Crossrefs

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

Sequence in context: A284152 A011024 A105855 * A079175 A332742 A202815

Adjacent sequences: A152951 A152952 A152953 * A152955 A152956 A152957

Keyword

sign

Author

Michael Somos, Dec 15 2008

Status

approved