A112146 McKay-Thompson series of class 9b for the Monster group.

1, 0, 9, -4, 0, 36, 2, 0, 126, 12, 0, 324, -21, 0, 801, 4, 0, 1764, 36, 0, 3744, -68, 0, 7452, 21, 0, 14400, 112, 0, 26748, -184, 0, 48510, 44, 0, 85536, 275, 0, 147924, -456, 0, 250452, 112, 0, 417276, 644, 0, 683640, -1019, 0, 1104948, 240, 0, 1761552, 1370, 0

Offset

-1,3

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

G. A. Edgar, Table of n, a(n) for n = -1..999

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

Expansion of q^(1/3) * 3*( b(q) / c(q) + c(q) / b(q)) in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Mar 24 2007

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u+v)^3 - (u^2 + 3*u - 18) * (v^2 + 3*v - 18).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + w^2 + u*w + 18*(u+w) - (w+u)*v^2 - 9*v + 54.

Expansion of ( (eta(q^3) / eta(q^9))^4 + 9 * (eta(q^9) / eta(q^3))^4) in powers of q.

a(3*n) = 0. a(3*n-1) = A058095(n). a(3*n + 1) = 9 * A128758(n). - Michael Somos, Feb 19 2015

Example

T9b = 1/q + 9*q - 4*q^2 + 36*q^4 + 2*q^5 + 126*q^7 + 12*q^8 + 324*q^10 + ...

Mathematica

QP = QPochhammer; s = (QP[q^3]^8 + 9*q^2*QP[q^9]^8)/(QP[q^3]^4*QP[q^9]^4) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^3 + A) / eta(x^9 + A))^4; polcoeff( A + 9*x^2 / A, n))}; /* Michael Somos, Mar 24 2007 */

Crossrefs

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

Cf. A058095, A128758.

Sequence in context: A393429 A354636 A021918 * A056897 A388079 A263192

Adjacent sequences: A112143 A112144 A112145 * A112147 A112148 A112149

Keyword

sign

Author

Michael Somos, Aug 28 2005, Aug 09 2008

Status

approved