A058095 McKay-Thompson series of class 9c for the Monster group.

1, -4, 2, 12, -21, 4, 36, -68, 21, 112, -184, 44, 275, -456, 112, 644, -1019, 240, 1370, -2156, 514, 2828, -4340, 992, 5498, -8392, 1930, 10428, -15675, 3528, 19060, -28472, 6399, 34072, -50382, 11184, 59333, -87260, 19312, 101496, -148148, 32480, 170130, -247156

Offset

0,2

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

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) in powers of q where b(), c() are cubic AGM theta functions.

Expansion of q^(1/3) * (eta(q) / eta(q^3))^4 in powers of q. - Michael Somos, Mar 24 2007

Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u+v)^3 - u*v * (u+3) * (v+3) .

Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*v^2 + v^2*w^2 - v*u^2*w^2 + u*w*v^2 - 9*u*w * (u+w).

G.f.: (Product_{k>0} (1 + x^k + x^(2*k)))^-4.

Euler transform of period 3 sequence [ -4, -4, 0, ...]. - Michael Somos, Mar 24 2007

a(n) = A112146(3*n - 1). Convolution inverse of A128758.

Example

G.f. = 1 - 4*x + 2*x^2 + 12*x^3 - 21*x^4 + 4*x^5 + 36*x^6 - 68*x^7 + ...
T9c = 1/q - 4*q^2 + 2*q^5 + 12*q^8 - 21*q^11 + 4*q^14 + 36*q^17 - 68*q^20 + ...

Mathematica

a[0] = 1; a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[(QPochhammer[x + A]/QPochhammer[x^3 + A])^4, n]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 06 2015, adapted from Michael Somos's PARI script *)

Prog

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

Crossrefs

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

Cf. A112146, A128758.

Sequence in context: A019239 A143944 A154345 * A105196 A372493 A167557

Adjacent sequences: A058092 A058093 A058094 * A058096 A058097 A058098

Keyword

sign

Author

N. J. A. Sloane, Nov 27 2000

Status

approved