A045490 McKay-Thompson series of class 8A for Monster.

1, 4, 36, 128, 386, 1024, 2488, 5632, 12031, 24576, 48308, 91904, 170110, 307200, 542872, 941056, 1602819, 2686976, 4439688, 7238272, 11657090, 18561024, 29242240, 45617664, 70507772, 108036096, 164192188, 247620352, 370726652, 551215104, 814216536, 1195226112, 1744133125, 2530738176

Offset

-1,2

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

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).

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

Index entries for McKay-Thompson series for Monster simple group

Formula

From Vaclav Kotesovec, Sep 07 2017: (Start)

a(n) = A007265(n) unless n=0.

a(n) ~ exp(sqrt(2*n)*Pi) / (2^(5/4) * n^(3/4)). (End)

Expansion of -4 + (eta(q^2)*eta(q^4)/(eta(q)*eta(q^8)))^8 in powers of q. - G. C. Greubel, Jun 02 2018

Mathematica

nmax = 50; CoefficientList[Series[-4*x + Product[((1 - x^(2*k))*(1 - x^(4*k))/((1 - x^k)*(1 - x^(8*k))))^8, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2017 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(-4 + (eta[q^2]*eta[q^4]/(eta[q]*eta[q^8]))^8), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 02 2018 *)

Prog

(PARI) q='q+O('q^30); a= -4 + (eta(q^2)*eta(q^4)/(eta(q)*eta(q^8)))^8/q; Vec(a) \\ G. C. Greubel, Jun 02 2018

Crossrefs

Cf. A007265.

Sequence in context: A076830 A144298 A072109 * A318150 A275133 A060783

Adjacent sequences: A045487 A045488 A045489 * A045491 A045492 A045493

Keyword

nonn

Author

N. J. A. Sloane

Status

approved