A112143 McKay-Thompson series of class 8D for the Monster group.

1, -4, 2, 8, -1, -20, -2, 40, 3, -72, 2, 128, -4, -220, -4, 360, 5, -576, 8, 904, -8, -1384, -10, 2088, 11, -3108, 12, 4552, -15, -6592, -18, 9448, 22, -13392, 26, 18816, -29, -26216, -34, 36224, 38, -49700, 42, 67728, -51, -91688, -56, 123392, 66, -165128, 78, 219784, -85, -291072

Offset: 0

Author

Michael Somos, Aug 28 2005

Keywords

• sign

Comments

The convolution square of this sequence is A007248, except for the constant term: T8D(q)^2 = T4C(q^2) - 8. - G. A. Edgar, Apr 02 2017

Examples

			T8D = 1/q -4*q +2*q^3 +8*q^5 -q^7 -20*q^9 -2*q^11 +40*q^13 +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..499 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. A007248, A046897.

Programs

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(1/2)*(eta[q]/eta[q^4])^4, {q, 0, 50}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, May 10 2018 *)

PARI

q='q+O('q^50); Vec((eta(q)/eta(q^4))^4) \\ G. C. Greubel, May 10 2018

Formula

Expansion of q^(1/2)*(eta(q) / eta(q^4))^4 in powers of q. - G. A. Edgar, Apr 02 2017

a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A046897(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 28 2017