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%I #25 Sep 07 2017 05:37:56
%S 1,10,51,204,681,1956,5135,12360,28119,60572,125682,251040,487426,
%T 920568,1699611,3070508,5445510,9490116,16283793,27537708,45959775,
%U 75760640,123471327,199081632,317814988
%N McKay-Thompson series of class 7A for the Monster group with a(0) = 10.
%H Seiichi Manyama, <a href="/A030183/b030183.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 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of Hauptmodul for X_0^{+}(7).
%F Expansion of (h + 7)^2 / h, where h = (eta(q) / eta(q^7))^4 in powers of q.
%F a(n) = A007264(n) = A045489(n) unless n = 0.
%F a(n) ~ exp(4*Pi*sqrt(n/7)) / (sqrt(2) * 7^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017
%e G.f. = 1/q + 10 + 51*q + 204*q^2 + 681*q^3 + 1956*q^4 + 5135*q^5 + 12360*q^6 + ...
%t a[ n_] := With[ {A = q (QPochhammer[ q^7] / QPochhammer[ q])^4}, SeriesCoefficient[ (1 + 7 A)^2 / A, {q, 0, n}]]; (* _Michael Somos_, Mar 30 2015 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^7 + A) / eta(x + A))^4; polcoeff( (1 + 7 * x * A)^2 / A, n))}; /* _Michael Somos_, Feb 02 2012 */
%Y Cf. A007264, A045489.
%K nonn
%O -1,2
%A _N. J. A. Sloane_.
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