A058486 McKay-Thompson series of class 12H for Monster.

1, 0, 14, 36, 85, 180, 360, 684, 1246, 2196, 3754, 6264, 10226, 16380, 25804, 40032, 61275, 92628, 138452, 204804, 300040, 435672, 627356, 896400, 1271525, 1791324, 2507426, 3488472, 4825531, 6638688, 9085888, 12373992, 16772908, 22633812

Offset

-1,3

Links

G. A. Edgar, Table of n, a(n) for n = -1..1002

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

Formula

Given g.f. A(x), then B(x) = A(x)+4 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = -u*v*(1 + u^2*v^2) + 7*u*v*(u + v)*(1 + u*v) + 9*u*v*(u^2 + v^2). - Michael Somos, May 16 2004

Expansion of (eta(q^3) * eta(q^4) / (eta(q) * eta(q^12)))^4 - 4 in powers of q. - Michael Somos, May 16 2004

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 11 2016

Example

T12H = 1/q + 14*q + 36*q^2 + 85*q^3 + 180*q^4 + 360*q^5 + 684*q^6 + ...

Mathematica

QP = QPochhammer; s = (QP[q^3]*(QP[q^4]/(QP[q]*QP[q^12])))^4 - 4*q + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 13 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^4 + A) / (eta(x + A) * eta(x^12 + A)))^4 - 4*x, n))}; /* Michael Somos, May 16 2004 */
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^6 + A)^6 / (eta(x + A)^5 * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)^5) - 5*x, n))}; /* Michael Somos, May 16 2004 */

Crossrefs

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

Sequence in context: A177722 A165761 A165765 * A279900 A263125 A113627

Adjacent sequences: A058483 A058484 A058485 * A058487 A058488 A058489

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Status

approved