A121666 McKay-Thompson series of class 6C for the Monster group with a(0) = -6.

1, -6, 15, -32, 87, -192, 343, -672, 1290, -2176, 3705, -6336, 10214, -16320, 25905, -39936, 61227, -92928, 138160, -204576, 300756, -435328, 626727, -897408, 1271205, -1790592, 2508783, -3487424, 4824825, -6641664, 9083400, -12371904, 16778784, -22630912

Offset

-1,2

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..147 from G. A. Edgar)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (1/q) * (chi(-q^3) * chi(-q))^6 in powers of q where chi() is a Ramanujan theta function.

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

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u,v) = v * u^2 + (12*v + 64) * u - v^2.

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

a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n/3)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Apr 09 2016

Example

T6C = 1/q - 6 + 15*q - 32*q^2 + 87*q^3 - 192*q^4 + 343*q^5 - 672*q^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q, q^2] QPochhammer[ q^3, q^6])^6, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q] QPochhammer[ q^3] / (QPochhammer[ q^2] QPochhammer[ q^6]))^6, {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^3 + A) / (eta(x^2 + A) * eta(x^6 + A)))^6, n))};
(PARI) N=66; q='q+O('q^N); Vec( ((eta(q^1)*eta(q^3))/(eta(q^2)*eta(q^6)))^6/q ) \\ Joerg Arndt, Apr 09 2016

Crossrefs

Cf. A007256, A045486.

Sequence in context: A273853 A192747 A231264 * A186829 A231452 A118734

Adjacent sequences: A121663 A121664 A121665 * A121667 A121668 A121669

Keyword

sign

Author

Michael Somos, Aug 14 2006

Status

approved