A045491 McKay-Thompson series of class 9A for the Monster group with a(0) = 3.

1, 3, 27, 86, 243, 594, 1370, 2916, 5967, 11586, 21870, 39852, 71052, 123444, 210654, 352480, 581013, 942786, 1510254, 2388204, 3734964, 5777788, 8852004, 13434984, 20218395, 30177684, 44704413, 65743348, 96033357, 139368816

Offset

-1,2

Comments

Given g.f. A(x), B(q) = 3 + A(q) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u + v)^3 + u*v*(27 + 9*(u + v) - u*v). - Michael Somos, Jun 16 2004

Expansion of eta(q^3)^12 / (eta(q) * eta(q^9))^6 - 3 in powers of q.

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra, Vol. 22, No. 13 (1994), 5175-5193.

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra, Vol. 18, No. 1 (1990), 253-278.

Index entries for McKay-Thompson series for Monster simple group

Formula

a(n) ~ exp(4*Pi*sqrt(n)/3) / (sqrt(6)*n^(3/4)). - Vaclav Kotesovec, May 01 2017

Example

G.f. = 1/q + 3 + 27*q + 86*q^2 + 243*q^3 + 594*q^4 + 1370*q^5 + 2916*q^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ -3 + (1/q) (QPochhammer[ q^3]^2 / (QPochhammer[ q] QPochhammer[ q^9]))^6, {q, 0, n}]; (* Michael Somos, Feb 22 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^3 + A)^12 / (eta(x + A) * eta(x^9 + A))^6 - 3*x, n))}; /* Michael Somos, Jun 16 2004 */

Crossrefs

Cf. A007266.

Sequence in context: A302725 A166102 A316754 * A318908 A343135 A343105

Adjacent sequences: A045488 A045489 A045490 * A045492 A045493 A045494

Keyword

nonn

Author

N. J. A. Sloane, Dec 11 1999

Status

approved