A134783 McKay-Thompson series of class 15A for the Monster group with a(0) = 1.

1, 1, 8, 22, 42, 70, 155, 246, 421, 722, 1101, 1730, 2761, 4062, 6106, 9040, 13065, 18806, 27081, 37950, 53183, 74290, 102213, 140048, 191612, 258426, 348300, 467484, 622023, 825016, 1090957, 1432290, 1875930, 2448610, 3179136, 4114996

Offset

-1,3

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

M. Koike, Mathieu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)

Index entries for McKay-Thompson series for Monster simple group

Formula

Associated with permutations in Mathieu group M24 of shape (15)(5)(3)(1).

G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A058508(n) unless n=0. Convolution with A030184 is A028998.

a(n) ~ exp(4*Pi*sqrt(n/15)) / (sqrt(2) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Expansion of A + 3 + 9/A, where A = (eta(q)*eta(q^5)/(eta(q^3)*eta(q^15)) ))^2, in powers of q. - G. C. Greubel, Jun 17 2018

Example

G.f. = 1/q + 1 + 8*q + 22*q^2 + 42*q^3 + 70*q^4 + 155*q^5 + 246*q^6 + 421*q^7 + ...

Mathematica

QP = QPochhammer; A = q^2*O[q]^40; A = (QP[q + A]*(QP[q^5 + A]/(QP[q^3 + A]*QP[q^15 + A])))^2/q; s = q*(3 + A + 9/A); CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)
a[ n_] := With[{A = (QPochhammer[ q^3] QPochhammer[ q^5] / (QPochhammer[ q] QPochhammer[ q^15]))^3 /q}, SeriesCoefficient[ -2 + A - 1/A, {q, 0, n}]]; (* Michael Somos, May 05 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, A = x^2 * O(x^n); A = (eta(x + A) * eta(x^5 + A) / (eta(x^3 + A) * eta(x^15 + A)))^2 / x; polcoeff( (3 + A + 9 / A), n))}

Crossrefs

Cf. A028998, A030184, A058498.

Sequence in context: A030999 A113744 A058508 * A211529 A069099 A172473

Adjacent sequences: A134780 A134781 A134782 * A134784 A134785 A134786

Keyword

nonn

Author

Michael Somos, Nov 22 2007

Status

approved