A121667 McKay-Thompson series of class 6D for the Monster group with a(0) = -4.

1, -4, -2, 28, -27, -52, 136, -108, -162, 620, -486, -760, 1970, -1404, -1940, 6048, -4293, -6100, 15796, -10692, -14264, 40232, -27108, -36496, 93285, -61020, -79054, 211624, -137781, -179296, 451680, -288360, -365780, 945836, -601020, -763016, 1897294, -1188756

Offset

-1,2

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 = -1..10000 (terms -1..148 from G. A. Edgar)

Formula

Expansion of 9 * b(q) * b(q^2) / (c(q) * c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.

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

Euler transform of period 6 sequence [ -4, -8, 0, -8, -4, 0, ...].

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

a(n) = A007257(n) = A045487(n) unless n=0. - Michael Somos, Feb 19 2015

Example

T6D = 1/q - 4 - 2*q + 28*q^2 - 27*q^3 - 52*q^4 + 136*q^5 - 108*q^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q] QPochhammer[ q^2] / (QPochhammer[ q^3] QPochhammer[ q^6]))^4, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)

Prog

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

Crossrefs

Cf. A045487, A007257.

Sequence in context: A286798 A123670 A200032 * A368767 A291844 A353750

Adjacent sequences: A121664 A121665 A121666 * A121668 A121669 A121670

Keyword

sign

Author

Michael Somos, Aug 14 2006

Status

approved