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%I #28 Mar 12 2021 22:24:43
%S 1,3,6,4,-3,-12,-8,12,30,20,-30,-72,-46,60,156,96,-117,-300,-188,228,
%T 552,344,-420,-1008,-603,732,1770,1048,-1245,-2976,-1776,2088,4908,
%U 2900,-3420,-7992,-4658,5460,12756,7408,-8583,-19944,-11564,13344,30756,17744,-20448,-46944
%N McKay-Thompson series of class 6E for the Monster group with a(0) = 3.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C Number 2 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - _Michael Somos_, Jul 21 2014
%C A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(6). [Yang 2004] - _Michael Somos_, Jul 21 2014
%H Seiichi Manyama, <a href="/A105559/b105559.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>
%H Y. Yang, <a href="http://dx.doi.org/10.1112/S0024609304003510">Transformation formulas for generalized Dedekind eta functions</a>, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of (eta(q^2) * eta(q^3)^3 / (eta(q) * eta(q^6)^3))^3 in powers of q.
%F G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = v^2 + 8*u + 6*u*v - u^2*v.
%F G.f.: x^-1 (Product_{k>0} (1 - x^(6*k - 3))^3 / (1 - x^(2*k - 1)))^3.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128643.
%F Expansion of (c(q) / c(q^2))^3 in powers of q where c() is a cubic AGM theta function.
%F Expansion of q^(-1) * (chi(-q^3)^3 / chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.
%F Euler transform of period 6 sequence [ 3, 0, -6, 0, 3, 0, ...].
%F a(n) = A007258(n) unless n=0. Convolution inverse of A123633.
%F Convolution cube of A062242. - _Michael Somos_, Apr 24 2015
%e G.f. = 1/q + 3 + 6*q + 4*q^2 - 3*q^3 - 12*q^4 - 8*q^5 + 12*q^6 + 30*q^7 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3, q^6]^3 / QPochhammer[ q, q^2])^3 / q, {q, 0, n}]; (* _Michael Somos_, Apr 24 2015 *)
%t a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 3, n, 6}] / Product[ 1 - q^k, {k, 1, n, 2}]^3)^3 / q, {q, 0, n}]; (* _Michael Somos_, Apr 24 2015 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( ( eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3) )^3, n))};
%Y Cf. A007258, A062242, A123633.
%K sign
%O -1,2
%A _Michael Somos_, Apr 13 2005, Jan 21 2009
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