A198955 q-expansion of modular form t_{3B}.

1, 15, 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, -65743304, -96033357, 278737632, -201031888, -288281592, 822239532, -583185916

Offset

-1,2

Comments

McKay-Thompson series of class 3B for the Monster group with a(0) = 15.

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

Links

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

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]

Formula

Expansion of 27 + (eta(q) / eta(q^3))^12 in powers of q. - Michael Somos, Nov 08 2011

Expansion of 27 * (a(q) / c(q))^3 in powers of q where a(q), c(q) are cubic AGM theta functions. - Michael Somos, Nov 08 2011

Convolution cube of A058091. Note psi_0 = a(q), psi_1 = c(q) where q = exp(2 Pi i tau). - Michael Somos, Nov 08 2011

Let psi_0 = theta_2(2*tau)*theta_2(6*tau) + theta_3(2*tau)*theta_3(6*tau),

psi_1 = (psi_0(tau/3) - psi_0(tau))/2;

then t_{3B} = 27(psi_0/psi_1)^3.

Example

1/q + 15 + 54*q - 76*q^2 - 243*q^3 + 1188*q^4 - 1384*q^5 - 2916*q^6 + 11934*q^7 - 11580*q^8 - 21870*q^9 + 79704*q^10 - 71022*q^11 - 123444*q^12 + 421308*q^13 - 352544*q^14 - 581013*q^15 + O(q^16)

Mathematica

a[ n_] :=  With[{m = n + 1}, SeriesCoefficient[ 27 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 = 27q+(QP[q]/QP[q^3])^12+O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 27*x + (eta(x + A) / eta(x^3 + A))^12, n))}; /* Michael Somos, Nov 08 2011 */

Crossrefs

Cf. A007244, A030182, A045481.

Sequence in context: A194454 A358398 A219384 * A341563 A063436 A010004

Adjacent sequences: A198952 A198953 A198954 * A198956 A198957 A198958

Keyword

sign

Author

N. J. A. Sloane, Oct 31 2011

Status

approved