A112204 McKay-Thompson series of class 63a for the Monster group.

1, 0, 2, 2, 1, 2, 3, 2, 4, 6, 6, 6, 9, 8, 13, 14, 15, 18, 23, 22, 29, 34, 35, 44, 52, 52, 65, 74, 80, 92, 110, 114, 134, 152, 164, 188, 215, 230, 266, 296, 321, 362, 412, 438, 503, 558, 602, 674, 755, 810, 912, 1010, 1093, 1210, 1346, 1446, 1614, 1772, 1922, 2118

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Index entries for McKay-Thompson series for Monster simple group

Formula

a(n) ~ exp(4*Pi*sqrt(n)/(3*sqrt(7))) / (sqrt(6) * 7^(1/4) * n^(3/4)). - Vaclav Kotesovec, May 30 2018

Expansion of (q*T21A)^(1/3), where T21A = A058563. - G. C. Greubel, Jun 20 2018

Expansion of q^(1/2)*(((eta(q^3)*eta(q^7))^2 - (eta(q)*eta(q^21))^2)/( eta(q)*eta(q^3)*eta(q^7)*eta(q^21)))^(2/3) in powers of q. - G. C. Greubel, Jun 20 2018

Example

T63a = 1/q + 2*q^5 + 2*q^8 + q^11 + 2*q^14 + 3*q^17 + 2*q^20 + 4*q^23 + ...

Mathematica

CoefficientList[Series[((QPochhammer[x^3]^2 * QPochhammer[x^7]^2 - x*QPochhammer[x]^2 * QPochhammer[x^21]^2) / (QPochhammer[x] * QPochhammer[x^3] * QPochhammer[x^7] * QPochhammer[x^21]))^(2/3), {x, 0, 100}], x] (* Vaclav Kotesovec, May 30 2018 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; nmax = 120; e21B := eta[q]*eta[q^3]/( eta[q^7]*eta[q^21]); T21A := 1 + e21B + 7/e21B; a:= CoefficientList[ Series[(q*T21A + O[q]^nmax)^(1/3), {q, 0, 60}], q]; Table[a[[n]], {n, 0, 50}] (* G. C. Greubel, Jun 20 2018 *)

Prog

(PARI) q='q+O('q^50); Vec((((eta(q^3)*eta(q^7))^2 - q*(eta(q)*eta(q^21) )^2)/(eta(q)*eta(q^3)*eta(q^7)*eta(q^21)))^(2/3)) \\ G. C. Greubel, Jun 20 2018

Crossrefs

Sequence in context: A029251 A263073 A133091 * A129710 A298857 A070680

Adjacent sequences: A112201 A112202 A112203 * A112205 A112206 A112207

Keyword

nonn

Author

Michael Somos, Aug 28 2005

Status

approved