A058489 McKay-Thompson series of class 12a for Monster.

1, -11, -21, -55, -110, -285, -450, -1001, -1605, -2936, -4740, -8271, -12571, -21075, -31755, -50104, -74600, -114816, -166570, -250601, -359499, -526106, -746512, -1074840, -1501836, -2131049, -2949216, -4117846, -5647010, -7795266, -10578308, -14451212, -19455813, -26294800, -35138260

Offset: 0

Author

N. J. A. Sloane, Nov 27 2000

Keywords

• sign

Comments

The convolution square of this sequence is A007254, except for the constant term: T12a(q)^2+22 = T6A(q^2). - G. A. Edgar, Apr 09 2017

Examples

			T12a = 1/q - 11*q - 21*q^3 - 55*q^5 - 110*q^7 - 285*q^9 - 450*q^11 - ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000 (terms 0..1005 from G. A. Edgar)
• D. Alexander, C. Cummins, J. McKay and C. Simons, Completely Replicable Functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
• D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
• Index entries for McKay-Thompson series for Monster simple group

Crossrefs

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[((1-x^k) * (1-x^(2*k)) / ((1-x^(3*k)) * (1-x^(6*k))))^2, {k, 1, nmax}] - 9*x*Product[((1-x^(3*k)) * (1-x^(6*k)) / ((1-x^k) * (1-x^(2*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2017 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; b := q^(1/2)*(eta[q]*eta[q^2]/(eta[q^3]*eta[q^6]))^2; a:= CoefficientList[Series[b - 9*q/b, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 13 2018 *)

PARI

q='q+O('q^30); A = (eta(q)*eta(q^2)/(eta(q^3)*eta(q^6)))^2; Vec(A - 9*q/A) \\ G. C. Greubel, Jun 13 2018

Formula

Expansion of q^(1/2) *( (eta(q)*eta(q^2)/(eta(q^3)*eta(q^6))^2 - 9*eta(q^3)*eta(q^6)/(eta(q)*eta(q^2)))^2 ) in powers of q. - G. A. Edgar, Apr 09 2017

a(n) ~ -exp(2*Pi*sqrt(n/3)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017