%I M5480 #37 Apr 01 2017 05:42:05

%S 1,0,783,8672,65367,371520,1741655,7161696,26567946,90521472,

%T 288078201,864924480,2469235686,6748494912,17746495281,45086909440,

%U 111066966315,266057139456,621284327856,1417338712800,3164665156308

%N McKay-Thompson series of class 3A for the Monster group with a(0) = 0.

%C Expansion of Hauptmodul for X_0^{+}(3).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Seiichi Manyama, <a href="/A007243/b007243.txt">Table of n, a(n) for n = -1..10000</a>

%H J. H. Conway and S. P. Norton, <a href="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.

%H N. D. Elkies, <a href="http://www.math.harvard.edu/~elkies/modular.pdf">Elliptic and modular curves over finite fields and related computational issues</a>, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 39.

%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 H. Strauss, <a href="http://dx.doi.org/10.1080/00927879008823911">The q-series of monstrous moonshine and the decomposition of the head characters</a>, Comm. Algebra 18 (1990), no. 1, 253-278.

%H Yang-Hui He, John McKay, <a href="http://arxiv.org/abs/1505.06742">Sporadic and Exceptional</a>, arXiv:1505.06742 [math.AG], 2015.

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F a(n) = A030197(n) = A045480(n) unless n = 0.

%F a(n) ~ exp(4*Pi*sqrt(n/3)) / (sqrt(2) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Apr 01 2017

%e T3A = 1/q + 783*q + 8672*q^2 + 65367*q^3 + 371520*q^4 + 1741655*q^5 + ...

%t QP = QPochhammer; A = q*O[q]^20; A = (QP[q^3+A]/QP[q+A])^12; s = (1+27*q* A)^2/A - 42*q; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 12 2015, adapted from PARI *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( (1 + 27 * x * A)^2 / A - 42 * x, n))} /* _Michael Somos_, Feb 02 2012 */

%Y Cf. A030197, A045480.

%K nonn,nice,easy

%O -1,3

%A _N. J. A. Sloane_