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A007246 McKay-Thompson series of class 2B for the Monster group.
(Formerly M5434)
9
1, 0, 276, -2048, 11202, -49152, 184024, -614400, 1881471, -5373952, 14478180, -37122048, 91231550, -216072192, 495248952, -1102430208, 2390434947, -5061476352, 10487167336, -21301241856, 42481784514, -83300614144 (list; graph; refs; listen; history; text; internal format)
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.

REFERENCES

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).

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.

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.

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.

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

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.

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-Fran├žois 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) = local(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: A028532 A028522 * A107080 A169976 A166192 A186466

Adjacent sequences:  A007243 A007244 A007245 * A007247 A007248 A007249

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified September 23 10:30 EDT 2014. Contains 247123 sequences.