A045481 McKay-Thompson series of class 3B for the Monster group with a(0) = -3.

1, -3, 54, -76, -243, 1188, -1384, -2916, 11934, -11580, -21870, 79704, -71022, -123444, 421308, -352544, -581013, 1885572, -1510236, -2388204, 7469928, -5777672, -8852004, 26869968, -20218587, -30177684, 89408826

Offset

-1,2

Links

Vaclav Kotesovec, 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.

N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 38.

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

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

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of 9 + (eta(q) / eta(q^3))^12 in powers of q.

Example

G.f. = 1/q - 3 + 54*q - 76*q^2 - 243*q^3 + 1188*q^4 - 1384*q^5 - 2916*q^6 + ...

Mathematica

a[ n_] :=  With[{m = n + 1}, SeriesCoefficient[ 9 q + (Product[ 1 - q^k, {k, m}] / Product[ 1 - q^k, {k, 3, m, 3}])^12, {q, 0, m}]] (* Michael Somos, Nov 08 2011 *)
QP = QPochhammer; s = 9*q+(QP[q]/QP[q^3])^12 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 9*x + (eta(x + A) / eta(x^3 + A))^12, n))}; /* Michael Somos, Nov 08 2011 */

Crossrefs

Essentially same as A007244, A030182, A045481.

Sequence in context: A093164 A092448 A344424 * A275566 A068380 A174782

Adjacent sequences: A045478 A045479 A045480 * A045482 A045483 A045484

Keyword

sign,easy,nice

Author

N. J. A. Sloane

Status

approved