A128633 McKay-Thompson series of class 6E for the Monster group with a(0) = 4.

1, 4, 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

Offset

-1,2

Comments

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

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

Expansion of q^-1 * (psi(q) / psi(q^3))^4 in powers of q where psi() is a Ramanujan theta function.

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

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (u - 9) * (u - 1) - (u - v)^2.

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

a(n) = A007258(n) = A105559(n) = A128632(n) unless n = 0.

Example

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

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q^(3/2)])^4, {q, 0, n}]; (* Michael Somos, Nov 12 2015 *)
QP = QPochhammer; s = (QP[q^2]^2*(QP[q^3]/(QP[q]*QP[q^6]^2)))^4 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015, adapted from PARI *)

Prog

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

Crossrefs

Cf. A007258, A105559, A128632.

Sequence in context: A199721 A349942 A187147 * A001482 A198493 A161965

Adjacent sequences: A128630 A128631 A128632 * A128634 A128635 A128636

Keyword

sign

Author

Michael Somos, Mar 15 2007

Status

approved