A187146 McKay-Thompson series of class 12B for the Monster group with a(0) = 5.

1, 5, 6, -4, -3, 12, -8, -12, 30, -20, -30, 72, -46, -60, 156, -96, -117, 300, -188, -228, 552, -344, -420, 1008, -603, -732, 1770, -1048, -1245, 2976, -1776, -2088, 4908, -2900, -3420, 7992, -4658, -5460, 12756, -7408, -8583, 19944, -11564, -13344, 30756

Offset

-1,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. A. Edgar, Table of n, a(n) for n = -1..1002

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (1/q) * (phi(q)^3 * psi(-q)) / (phi(q^3) * psi(-q^3)^3) in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of eta(q^2)^14 / (eta(q)^5 * eta(q^3) * eta(q^4)^5 * eta(q^6)^2 * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ 5, -9, 6, -4, 5, -6, 5, -4, 6, -9, 5, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 9 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A164617.

Convolution of A113660 and A133637.

a(n) = -(-1)^n * A128632(n). - Michael Somos, May 20 2015

Example

G.f. = 1/q + 5 + 6*q - 4*q^2 - 3*q^3 + 12*q^4 - 8*q^5 - 12*q^6 + 30*q^7 - 20*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 3, 0, q]^3 EllipticTheta[ 2, Pi/4, q^(1/2)]  / (EllipticTheta[ 3, 0, q^3] EllipticTheta[ 2, Pi/4, q^(3/2)]^3), {q, 0, n}]; (* Michael Somos, May 20 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^14 / (eta(x + A)^5 * eta(x^3 + A) * eta(x^4 + A)^5 * eta(x^6 + A)^2 * eta(x^12 + A)), n))};

Crossrefs

Cf. A112148, A113660, A128632, A133637, A164617.

Sequence in context: A221216 A229073 A019818 * A128632 A197490 A319015

Adjacent sequences: A187143 A187144 A187145 * A187147 A187148 A187149

Keyword

sign

Author

Michael Somos, Mar 05 2011

Status

approved