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%I #32 Feb 14 2021 01:15:33
%S 1,15,54,-76,-243,1188,-1384,-2916,11934,-11580,-21870,79704,-71022,
%T -123444,421308,-352544,-581013,1885572,-1510236,-2388204,7469928,
%U -5777672,-8852004,26869968,-20218587,-30177684,89408826,-65743304,-96033357,278737632,-201031888,-288281592,822239532,-583185916
%N q-expansion of modular form t_{3B}.
%C McKay-Thompson series of class 3B for the Monster group with a(0) = 15.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A198955/b198955.txt">Table of n, a(n) for n = -1..1000</a>
%H Masao Koike, <a href="https://oeis.org/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
%F Expansion of 27 + (eta(q) / eta(q^3))^12 in powers of q. - _Michael Somos_, Nov 08 2011
%F Expansion of 27 * (a(q) / c(q))^3 in powers of q where a(q), c(q) are cubic AGM theta functions. - _Michael Somos_, Nov 08 2011
%F Convolution cube of A058091. Note psi_0 = a(q), psi_1 = c(q) where q = exp(2 Pi i tau). - _Michael Somos_, Nov 08 2011
%F Let psi_0 = theta_2(2*tau)*theta_2(6*tau) + theta_3(2*tau)*theta_3(6*tau),
%F psi_1 = (psi_0(tau/3) - psi_0(tau))/2;
%F then t_{3B} = 27(psi_0/psi_1)^3.
%e 1/q + 15 + 54*q - 76*q^2 - 243*q^3 + 1188*q^4 - 1384*q^5 - 2916*q^6 + 11934*q^7 - 11580*q^8 - 21870*q^9 + 79704*q^10 - 71022*q^11 - 123444*q^12 + 421308*q^13 - 352544*q^14 - 581013*q^15 + O(q^16)
%t a[ n_] := With[{m = n + 1}, SeriesCoefficient[ 27 q + (Product[ 1 - q^k, {k, m}] / Product[ 1 - q^k, {k, 3, m, 3}])^12, {q, 0, m}]]; (* _Michael Somos_, Nov 08 2011 *)
%t QP = QPochhammer; s = 27q+(QP[q]/QP[q^3])^12+O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 16 2015, adapted from PARI *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 27*x + (eta(x + A) / eta(x^3 + A))^12, n))}; /* _Michael Somos_, Nov 08 2011 */
%Y Cf. A007244, A030182, A045481.
%K sign
%O -1,2
%A _N. J. A. Sloane_, Oct 31 2011
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