A101558 McKay-Thompson series of class 2A for the Monster group.

1, 0, 4372, 96256, 1240002, 10698752, 74428120, 431529984, 2206741887, 10117578752, 42616961892, 166564106240, 611800208702, 2125795885056, 7040425608760, 22327393665024, 68134255043715, 200740384538624

Offset

-1,3

Comments

Hauptmodul for Gamma_0(2)+.

References

T. Gannon, Moonshine Beyond the Monster, Cambridge, 2006; see p. 423.

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

R. E. Borcherds, Review of "Moonshine Beyond the Monster ..." (Cambridge, 2006), Bull. Amer. Math. Soc., 45 (2008), 675-679.

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Index entries for McKay-Thompson series for Monster simple group

Formula

a(n) ~ exp(2*Pi*sqrt(2*n)) / (2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Apr 01 2017

Example

T2A = 1/q + 4372q + 96256q^2 + 1240002q^3 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; f2A:= (eta[q]/eta[q^2])^24*(1 + 64*( eta[q^2]/eta[q])^24)^2; a:= CoefficientList[Series[q*(f2A - 104), {q, 0, 50}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, May 10 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, A = prod(k=1, n\2+1, 1 - x^(2*k-1), 1 + x^2 * O(x^n))^24; polcoeff(64^2*x/A + A/x + 24, n))};

Crossrefs

A045478, A007241, A106207, A007267, A101558 are all essentially the same sequence.

Cf. A007241 (same except for 0th term), A007267, A045478.

Sequence in context: A207047 A001378 A028511 * A163583 A203403 A206148

Adjacent sequences: A101555 A101556 A101557 * A101559 A101560 A101561

Keyword

nonn

Author

Michael Somos, Dec 06 2004

Status

approved