A128632 McKay-Thompson series of class 6E 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, 17744, -20448, -46944

Offset

-1,2

Comments

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

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of 3 * (b(q)^2 / b(q^2)) / (c(q^2)^2 / c(q)) in powers of q where b(), c() are cubic AGM theta functions.

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

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

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (u + 8) * (v + 9) - (u - v)^2.

G.f.: (1/x) * (Product_{k>0} (1 + x^k) * (1 + x^(3*k)) * ((1 - x^(6*k)) / (1 - x^k))^4)^-1.

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 72 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128638.

a(n) = A007258(n) = A045488(n) = A105559(n) = A128633(n) unless n = 0. Convolution inverse of A128638.

a(n) = -(-1)^n * A187146(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 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 4 EllipticTheta[ 4, 0, q]^3 EllipticTheta[ 2, 0, q^(1/2)] / (EllipticTheta[ 4, 0, q^3] EllipticTheta[ 2, 0, q(3/2)]^3), {q, 0, n}]; (* Michael Somos, May 20 2015 *)
QP = QPochhammer; s = (QP[q]/QP[q^6])^5*(QP[q^3]/QP[q^2]) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)

Prog

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

Crossrefs

Cf. A007258, A045488, A105559, A128633, A128638, A187146, A258094.

Sequence in context: A229073 A019818 A187146 * A197490 A319015 A229481

Adjacent sequences: A128629 A128630 A128631 * A128633 A128634 A128635

Keyword

sign

Author

Michael Somos, Mar 15 2007

Status

approved