A112180 McKay-Thompson series of class 40a for the Monster group.

1, 0, 3, 4, 4, 4, 7, 12, 13, 16, 22, 28, 38, 44, 55, 72, 83, 104, 129, 156, 187, 220, 273, 328, 384, 452, 539, 652, 757, 880, 1041, 1220, 1428, 1652, 1924, 2244, 2585, 2992, 3458, 3992, 4581, 5244, 6053, 6936, 7910, 9024, 10303, 11784, 13380, 15176

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

Expansion of A - q/A, where A = q^(1/2)*(eta(q^4)*eta(q^5)/( eta(q)* eta(q^20))), in powers of q. - G. C. Greubel, Jun 26 2018

a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(5/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 27 2018

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (8/25) * exp(-Pi / 2) * 2^(2/5) * 5^(3/4) * Gamma(9/10)^2 * Gamma(7/10)^2 * (5+5^(1/2))^2 * (1/4*5^(1/2)-1/4)^2 * (1/4*5^(1/2)+1/4)^2 / Gamma(4/5)^4 = A388522. - Simon Plouffe, Sep 17 2025

Example

T40a = 1/q +3*q^3 +4*q^5 +4*q^7 +4*q^9 +7*q^11 +12*q^13 +...

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; A:= q^(1/2)*(eta[q^4]*eta[q^5]/( eta[q]*eta[q^20])); a := CoefficientList[Series[A - q/A, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 26 2018 *)

Prog

(PARI) q='q+O('q^50); A = eta(q^4)*eta(q^5)/(eta(q)*eta(q^20)); Vec(A - q/A) \\ G. C. Greubel, Jun 26 2018

Crossrefs

Sequence in context: A259884 A283972 A213509 * A058559 A232092 A345196

Adjacent sequences: A112177 A112178 A112179 * A112181 A112182 A112183

Keyword

nonn,changed

Author

Michael Somos, Aug 28 2005

Status

approved