|
|
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
|
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
|
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.
G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (pdf, ps).
|
|
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
|
|
|
KEYWORD
|
sign,easy,nice
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|