A121666 - OEIS

%I #23 Mar 12 2021 22:24:44

%S 1,-6,15,-32,87,-192,343,-672,1290,-2176,3705,-6336,10214,-16320,

%T 25905,-39936,61227,-92928,138160,-204576,300756,-435328,626727,

%U -897408,1271205,-1790592,2508783,-3487424,4824825,-6641664,9083400,-12371904,16778784,-22630912

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

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

%H Seiichi Manyama, <a href="/A121666/b121666.txt">Table of n, a(n) for n = -1..10000</a> (terms -1..147 from G. A. Edgar)

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

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

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

%F 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.

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

%F a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n/3)) / (2*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Apr 09 2016

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

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q, q^2] QPochhammer[ q^3, q^6])^6, {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%t 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 *)

%o (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))};

%o (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

%Y Cf. A007256, A045486.

%K sign

%O -1,2

%A _Michael Somos_, Aug 14 2006