A121667 - OEIS

%I #17 Mar 30 2017 08:21:26

%S 1,-4,-2,28,-27,-52,136,-108,-162,620,-486,-760,1970,-1404,-1940,6048,

%T -4293,-6100,15796,-10692,-14264,40232,-27108,-36496,93285,-61020,

%U -79054,211624,-137781,-179296,451680,-288360,-365780,945836,-601020,-763016,1897294,-1188756

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

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H Seiichi Manyama, <a href="/A121667/b121667.txt">Table of n, a(n) for n = -1..10000</a> (terms -1..148 from G. A. Edgar)

%F 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.

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

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

%F 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)).

%F a(n) = A007257(n) = A045487(n) unless n=0. - _Michael Somos_, Feb 19 2015

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

%t 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 *)

%o (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))};

%Y Cf. A045487, A007257.

%K sign

%O -1,2

%A _Michael Somos_, Aug 14 2006