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%I #30 Mar 06 2019 16:28:58
%S 1,3,27,86,243,594,1370,2916,5967,11586,21870,39852,71052,123444,
%T 210654,352480,581013,942786,1510254,2388204,3734964,5777788,8852004,
%U 13434984,20218395,30177684,44704413,65743348,96033357,139368816
%N McKay-Thompson series of class 9A for the Monster group with a(0) = 3.
%C Given g.f. A(x), B(q) = 3 + A(q) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u + v)^3 + u*v*(27 + 9*(u + v) - u*v). - _Michael Somos_, Jun 16 2004
%C Expansion of eta(q^3)^12 / (eta(q) * eta(q^9))^6 - 3 in powers of q.
%H G. C. Greubel, <a href="/A045491/b045491.txt">Table of n, a(n) for n = -1..1000</a>
%H J. H. Conway and S. P. Norton, <a href="https://doi.org/10.1112/blms/11.3.308">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.
%H D. Ford, J. McKay and S. P. Norton, <a href="https://doi.org/10.1080/00927879408825127">More on replicable functions</a>, Comm. Algebra, Vol. 22, No. 13 (1994), 5175-5193.
%H J. McKay and H. Strauss, <a href="https://doi.org/10.1080/00927879008823911">The q-series of monstrous moonshine and the decomposition of the head characters</a>, Comm. Algebra, Vol. 18, No. 1 (1990), 253-278.
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F a(n) ~ exp(4*Pi*sqrt(n)/3) / (sqrt(6)*n^(3/4)). - _Vaclav Kotesovec_, May 01 2017
%e G.f. = 1/q + 3 + 27*q + 86*q^2 + 243*q^3 + 594*q^4 + 1370*q^5 + 2916*q^6 + ...
%t a[ n_] := SeriesCoefficient[ -3 + (1/q) (QPochhammer[ q^3]^2 / (QPochhammer[ q] QPochhammer[ q^9]))^6, {q, 0, n}]; (* _Michael Somos_, Feb 22 2015 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^3 + A)^12 / (eta(x + A) * eta(x^9 + A))^6 - 3*x, n))}; /* _Michael Somos_, Jun 16 2004 */
%Y Cf. A007266.
%K nonn
%O -1,2
%A _N. J. A. Sloane_, Dec 11 1999
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