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A134782 McKay-Thompson series of class 14A for the Monster group with a(0) = 1. 3
1, 1, 11, 20, 57, 92, 207, 312, 623, 932, 1674, 2464, 4162, 6024, 9595, 13748, 21126, 29820, 44449, 62004, 90191, 124288, 177135, 241632, 338508, 457272, 631031, 845008, 1150752, 1528380, 2057700, 2712192, 3614217, 4730148, 6245541, 8119672 (list; graph; refs; listen; history; text; internal format)
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 (14)(7)(2)(1).

G.f. is a period 1 Fourier series which satisfies f(-1 / (14 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A058497(n) unless n=0. Convolution with A030187 is A028997.

a(n) ~ exp(2*Pi*sqrt(2*n/7)) / (2^(3/4) * 7^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Expansion of A - 3 + 1/A, where A = (eta(q^2)*eta(q^7)/(eta(q)*eta(q^14) ))^4, in powers of q. - G. C. Greubel, Jun 17 2018

EXAMPLE

G.f. = 1/q + 1 + 11*q + 20*q^2 + 57*q^3 + 92*q^4 + 207*q^5 + 312*q^6 + 623*q^7 + ...

MATHEMATICA

QP = QPochhammer; A = q^2*O[q]^40; A = (QP[q + A]*(QP[q^7 + A]/(QP[q^2 + A]*QP[q^14 + A])))^3/q; s = q*(4 + A + 8/A); CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 15 2015, adapted from PARI *)

a[ n_] := With[{A = (QPochhammer[ q] QPochhammer[ q^7] / (QPochhammer[ q^2] QPochhammer[ q^14]))^3 / q}, SeriesCoefficient[ 4 + A + 8 / A, {q, 0, n}]]; (* Michael Somos, May 05 2016 *)

eta[q_]:= q^(1/24)*QPochhammer[q]; A:= (eta[q^2]*eta[q^7]/(eta[q]* eta[q^14]))^4; a:= CoefficientList[Series[q*(A - 3 + 1/A), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 17 2018 *)

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, A = x^2 * O(x^n); A = (eta(x + A) * eta(x^7 + A) / (eta(x^2 + A) * eta(x^14 + A)))^3 / x; polcoeff( (4 + A + 8 / A), n))};

(PARI) q='q+O('q^50); A = (eta(q^2)*eta(q^7)/(eta(q)*eta(q^14)))^4/q; Vec(A - 3 +1/A) \\ G. C. Greubel, Jun 17 2018

CROSSREFS

Cf. A028997, A030187, A058497.

Sequence in context: A076851 A164576 A058497 * A067969 A068599 A180113

Adjacent sequences:  A134779 A134780 A134781 * A134783 A134784 A134785

KEYWORD

nonn

AUTHOR

Michael Somos, Nov 22 2007

STATUS

approved

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Last modified January 22 16:47 EST 2019. Contains 319364 sequences. (Running on oeis4.)