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%I #22 Jan 25 2018 02:55:18
%S 1,5,-7,3,15,-32,9,58,-96,22,149,-253,68,372,-599,140,826,-1317,317,
%T 1768,-2735,632,3526,-5434,1259,6854,-10364,2346,12765,-19188,4345,
%U 23224,-34524,7693,41049,-60654,13487,71176,-104303,22962,120718,-176050,38622,201539
%N McKay-Thompson series of class 9B for the Monster group.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A058091/b058091.txt">Table of n, a(n) for n = 0..1000</a>
%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H J. McKay and A. Sebbar, <a href="http://dx.doi.org/10.1007/s002080000116">Fuchsian groups, automorphic functions and Schwarzians</a>, Math. Ann., 318 (2000), 255-275.
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of 3 * q^(1/3) * a(q) / c(q) in powers of q where a(), c() are cubic AGM theta functions. - _Michael Somos_, Aug 09 2006
%F Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^3 + v^3 - u^2*v^2 + 9*u*v - 54. - _Michael Somos_, Aug 09 2006
%F Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^3 + 1) * (v^3 + 1) - (u*v + 5) * (u^2*v^2 - 4*u*v + 11). - _Michael Somos_, Aug 20 2014
%F G.f.: Sum_{k>=0} a(k) * x^(3*k) = 3*x + (Product_{k>0} (1 - x^k) / (1 - x^(9*k)))^3. - _Michael Somos_, Aug 09 2006
%F a(n) = A131986(3*n - 1). - _Michael Somos_, Aug 20 2014
%e G.f. = 1 + 5*x - 7*x^2 + 3*x^3 + 15*x^4 - 32*x^5 + 9*x^6 + 58*x^7 - 96*x^8 + ...
%e T9B = 1/q + 5*q^2 - 7*q^5 + 3*q^8 + 15*q^11 - 32*q^14 + 9*q^17 + 58*q^20 + ...
%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q] / QPochhammer[ q^9])^3 + 3, {q, 0, 3 n - 1}]; (* _Michael Somos_, Aug 20 2014 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, n*=3; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^9 + A))^3, n))}; /* _Michael Somos_, Aug 09 2006 */
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = eta(x + A) / eta(x^2 + A); A = subst(A + x * O(x^(n\3)), x, x^3)^3 / A; polcoeff( A + 4*x / A^2, n))}; /* _Michael Somos_, Aug 09 2006 */
%Y Cf. A007248, A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y Cf. A131986.
%K sign,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 27 2000
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