|
|
A045479
|
|
McKay-Thompson series of class 2B for the Monster group with a(0) = -8.
|
|
7
|
|
|
1, -8, 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
|
-1,2
|
|
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
|
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 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 *)
|
|
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
|
|
|
KEYWORD
|
sign,easy,nice
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|