A058096 McKay-Thompson series of class 9d for Monster.

1, 0, -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

Offset

-1,3

Links

G. C. Greubel, Table of n, a(n) for n = -1..5000 (terms -1..997 from G. A. Edgar)

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

Index entries for McKay-Thompson series for Monster simple group

Formula

G.f. is a period 1 Fourier series which satisfies f(-1 / (81 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 28 2015

a(3*n - 1) = A058601(n). a(3*n) = 0. a(3*n + 1) = -3 * A192309(n). - Michael Somos, Aug 28 2015

Expansion of A - 3/A, where A = (eta(q^9)^2/(eta(q^3)*eta(q^27)))^2, in powers of q. - G. C. Greubel, Jun 03 2018

Example

T9d = 1/q - 3*q + 2*q^2 + 6*q^4 + 5*q^5 + 3*q^7 + 6*q^8 - 18*q^10 + 12*q^11 + ...

Mathematica

a[ n_] := With[ {A = 1/q (QPochhammer[ q^9]^2 / (QPochhammer[ q^3] QPochhammer[ q^27]))^2}, seriesCoefficient[ A - 3 / A, {q, 0, n}]]; (* Michael Somos, Aug 28 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; A := (eta[q^9]^2/(eta[q^3]*eta[q^27]) )^2; a := CoefficientList[Series[q*(A - 3/A), {q, 0, 60}], q];
Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 03 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = eta(x^9 + A)^4 / (eta(x^3 + A) * eta(x^27 + A))^2; polcoeff( A - 3 * x^2 / A, n))}; /* Michael Somos, Aug 28 2015 */

Crossrefs

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

Cf. A058601, A192309.

Sequence in context: A079408 A114376 A216395 * A049780 A159584 A257653

Adjacent sequences: A058093 A058094 A058095 * A058097 A058098 A058099

Keyword

sign

Author

N. J. A. Sloane, Nov 27 2000

Status

approved