A007246 McKay-Thompson series of class 2B for the Monster group. (Formerly M5434)

1, 0, 276, -2048, 11202, -49152, 184024, -614400, 1881471, -5373952, 14478180, -37122048, 91231550, -216072192, 495248952, -1102430208, 2390434947, -5061476352, 10487167336, -21301241856, 42481784514, -83300614144

Offset

-1,3

Comments

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

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

T. Gannon, Moonshine Beyond the Monster, Cambridge, 2006; see pp. 139, 424.

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

Michael Somos, Introduction to Ramanujan theta functions

Index entries for sequences related to groups

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of 24 + chi(-q)^24 / q in powers of q where chi() is a Ramanujan theta function.

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

Example

T2B = 1/q + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 - ...

Mathematica

a[0] = 0; 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 *)
a[ n_] := SeriesCoefficient[ 24 + 1/q QPochhammer[ q, q^2]^24, {q, 0, n}]; (* Michael Somos, Jul 05 2014 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 24 * x + (eta(x + A) / eta(x^2 + A))^24, n))}; /* Michael Somos, Jul 05 2014 */

Crossrefs

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

Sequence in context: A128382 A028532 A028522 * A107080 A333049 A297525

Adjacent sequences: A007243 A007244 A007245 * A007247 A007248 A007249

Keyword

sign,easy,nice

Author

N. J. A. Sloane

Status

approved