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A107080 McKay-Thompson series of class 4A for the Monster group. 8
1, 0, 276, 2048, 11202, 49152, 184024, 614400, 1881471, 5373952, 14478180, 37122048, 91231550, 216072192, 495248952, 1102430208, 2390434947, 5061476352, 10487167336, 21301241856, 42481784514, 83300614144, 160791890304, 305854488576, 573872089212, 1063005978624, 1945403602764, 3519965179904 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,3

COMMENTS

Also character of extremal vertex operator algebra of rank 12.

LINKS

Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..1000 from T. D. Noe)

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).

G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, arXiv:0706.0236 [math.QA], 2007, from Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.

Index entries for McKay-Thompson series for Monster simple group

FORMULA

G.f.: (1/x)(Product_{k>0} (1+x^k)/(1+x^(2k)))^24 -24.

a(n) ~ exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - Vaclav Kotesovec, Sep 06 2015

EXAMPLE

T4A = 1/q + 276q + 2048q^2 + 11202q^3 + 49152q^4 + 184024q^5 +...

MATHEMATICA

a[0] = 0; a[n_] := SeriesCoefficient[ Product[1 - q^k, {k, 1, n+1, 2}]^24/q, {q, 0, n}] // Abs; Table[a[n], {n, -1, 20}] (* Jean-François Alcover, Oct 14 2013, after Michael Somos *)

QP = QPochhammer; s = (QP[q^2]^2/QP[q]/QP[q^4])^24 - 24*q + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, after Michael Somos *)

PROG

(PARI) {a(n)=local(A); if(n<-1, 0, n++; A=x*O(x^n); polcoeff( (eta(x^2+A)^2/eta(x+A)/eta(x^4+A))^24-24*x, n))}

(PARI) q='q+O('q^66); Vec(+40*q+(eta(q)^4 / eta(q^4)^4 - q*4^2*eta(q^4)^4 / eta(q)^4)^2) \\ Joerg Arndt, Mar 23 2017

CROSSREFS

Cf. A007246(n)=-(-1)^n a(n).

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

Sequence in context: A028532 A028522 A007246 * A169976 A166192 A186466

Adjacent sequences:  A107077 A107078 A107079 * A107081 A107082 A107083

KEYWORD

nonn

AUTHOR

Michael Somos, May 11 2005

STATUS

approved

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Last modified December 13 09:55 EST 2017. Contains 295957 sequences.