A112175 McKay-Thompson series of class 36e for the Monster group.

1, -1, 0, -2, 2, -1, 2, -2, 3, -4, 4, -4, 7, -7, 6, -10, 11, -11, 14, -16, 17, -21, 22, -24, 32, -34, 34, -44, 49, -50, 60, -66, 72, -84, 90, -98, 117, -125, 132, -156, 171, -181, 206, -226, 245, -277, 298, -322, 369, -397, 422, -480, 522, -557, 620, -674, 728, -807, 868, -936, 1043, -1121, 1198

Offset: 0

Author

Michael Somos, Aug 28 2005

Keywords

• sign

Examples

			T36e = 1/q - q^5 - 2*q^17 + 2*q^23 - q^29 + 2*q^35 - 2*q^41 + 3*q^47 + ...
		

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

Crossrefs

Cf. A112206.

Programs

Mathematica

nmax = 60; CoefficientList[Series[1/Product[(1 + x^(3*k))*(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 06 2018 *)
eta[q_] := q^(1/24)*QPochhammer[q];  a:= CoefficientList[Series[q^(1/6)*(eta[q]*eta[q^3]/(eta[q^2]*eta[q^6])), {q, 0, 60}], q]; Table[a[[n]], {n, 0, 50}] (* G. C. Greubel, Jun 19 2018 *)

PARI

q='q+O('q^60); Vec(eta(q)*eta(q^3)/(eta(q^2)*eta(q^6))) \\ G. C. Greubel, Jun 19 2018

Formula

a(n) = (-1)^n * A112206(n). - Vaclav Kotesovec, Jun 06 2018

a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi/3) / (2^(5/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

Expansion of q^(1/6)*eta(q)*eta(q^3)/(eta(q^2)*eta(q^6)) in powers of q. - G. C. Greubel, Jun 19 2018