A045479 McKay-Thompson series of class 2B for the Monster group with a(0) = -8.

1, -8, 276, -2048, 11202, -49152, 184024, -614400, 1881471, -5373952, 14478180, -37122048, 91231550, -216072192, 495248952, -1102430208, 2390434947, -5061476352, 10487167336, -21301241856, 42481784514, -83300614144

Offset

-1,2

Comments

Unsigned sequence gives McKay-Thompson series of class 4A for Monster; also character of extremal vertex operator algebra of rank 12.

The value of a(0) is the Rademacher constant for the modular function and appears in Conway and Norton's Table 4. - Michael Somos, Mar 08 2011

References

G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.

Links

T. D. Noe, Table of n, a(n) for n = -1..1000

R. E. Borcherds, Introduction to the monster Lie algebra, pp. 99-107 of M. Liebeck and J. Saxl, editors, Groups, Combinatorics and Geometry (Durham, 1990). London Math. Soc. Lect. Notes 165, Cambridge Univ. Press, 1992.

B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

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

G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (pdf, ps).

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of 16 + (eta(q) / eta(q^2))^24 in powers of q. - Michael Somos, Mar 08 2011

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

Example

1/q - 8 + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 + ...

Mathematica

a[0] = -8; a[n_] := SeriesCoefficient[ Product[1 - q^k, {k, 1, n+1, 2}]^24/q, {q, 0, n}]; Table[a[n], {n, -1, 20}] (* Jean-François Alcover, Oct 14 2013, after Michael Somos *)
QP = QPochhammer; s = 16*q + (QP[q]/QP[q^2])^24 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, after Michael Somos *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 16 * x + (eta(x + A) / eta(x^2 + A))^24, n))}; /* Michael Somos, Mar 08 2011 */

Crossrefs

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

Sequence in context: A338636 A335586 A159496 * A179570 A201188 A296411

Adjacent sequences: A045476 A045477 A045478 * A045480 A045481 A045482

Keyword

sign,easy,nice

Author

N. J. A. Sloane

Status

approved