A034318 McKay-Thompson series of class 13A for the Monster group with a(0) = -2.

1, -2, 12, 28, 66, 132, 258, 468, 843, 1428, 2406, 3900, 6253, 9780, 15144, 22980, 34599, 51300, 75430, 109584, 158052, 225676, 320082, 450216, 629329, 873444, 1205514, 1653364, 2256087, 3061620, 4135280, 5557980, 7438170, 9910132, 13151568, 17382756, 22891391

Offset

-1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = -1..5000

I. Chen and N. Yui, Singular values of Thompson series. In Groups, difference sets and the Monster (Columbus, OH, 1993), pp. 255-326, Ohio State University Mathematics Research Institute Publications, 4, de Gruyter, Berlin, 1996.

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).

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of Hauptmodul for Gamma_0(13)+.

Expansion of (eta(q) / eta(q^13))^2 + 13*(eta(q^13) / eta(q))^2 in powers of q. - Michael Somos, Jul 05 2012

G.f.: (1/x) * Product_{k>0} ((1 - x^k) / (1 - x^(13*k)))^2 + 13*x * Product_{k>0} ((1 - x^(13*k)) / (1 - x^k))^2. - Michael Somos, Jul 05 2012

a(n) = A034319(n) unless n=0. - Michael Somos, Jul 05 2012

a(n) = A133099(n) + 13 * A121597(n). - Michael Somos, Jul 05 2012

a(n) ~ exp(4*Pi*sqrt(n/13)) / (sqrt(2) * 13^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2017

Example

G.f. = 1/q - 2 + 12*q + 28*q^2 + 66*q^3 + 132*q^4 + 258*q^5 + 468*q^6 + ...

Mathematica

QP = QPochhammer; A = O[q]^40; A = (QP[q + A]/QP[q^13 + A])^2; s = A + 13*(q^2/A); CoefficientList[s, q] (* Jean-François Alcover, Nov 13 2015, adapted from PARI *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x + A) / eta(x^13 + A))^2; polcoeff( A + 13 * x^2 / A, n))}; /* Michael Somos, Jul 05 2012 */

Crossrefs

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

Cf. A034319, A121597, A133099.

Sequence in context: A164876 A225291 A211623 * A338798 A345694 A326517

Adjacent sequences: A034315 A034316 A034317 * A034319 A034320 A034321

Keyword

sign,easy

Author

N. J. A. Sloane

Status

approved