A134786 McKay-Thompson series of class 4A for the Monster group with a(0) = 4.

1, 4, 276, 2048, 11202, 49152, 184024, 614400, 1881471, 5373952, 14478180, 37122048, 91231550, 216072192, 495248952, 1102430208, 2390434947, 5061476352, 10487167336, 21301241856, 42481784514, 83300614144, 160791890304

Offset

-1,2

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 (4)^4(2)^2(1)^4.

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A107080(n) unless n=0. Convolution with A030212 is A037219.

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

Example

G.f. = 1/q + 4 + 276*q + 2048*q^2 + 11202*q^3 + 49152*q^4 + 184024*q^5 + ...

Mathematica

a[0] = 4; a[n_] := SeriesCoefficient[ Product[1 - q^k, {k, 1, n+1, 2}]^24/q, {q, 0, n}] // Abs; Table[a[n], {n, -1, 21}] (* Jean-François Alcover, Oct 14 2013, after Michael Somos *)
QP = QPochhammer; s = (QP[q^2]^2/QP[q]/QP[q^4])^24 - 20*q + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, after Michael Somos *)
a[ n_] := SeriesCoefficient[ -20 + QPochhammer[ -q, q^2]^24 / q, {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^4 + A))^8 / x; polcoeff( 12 + A + 256 / A, n))};

Crossrefs

Cf. A030212, A037219, A107080.

Cf. A097340. [From R. J. Mathar, Dec 13 2008]

A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.

Sequence in context: A340916 A000320 A101758 * A290225 A190635 A202031

Adjacent sequences: A134783 A134784 A134785 * A134787 A134788 A134789

Keyword

nonn

Author

Michael Somos, Nov 22 2007

Status

approved