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%I #50 Dec 24 2017 16:06:14
%S 1,0,196884,21493760,864299970,20245856256,333202640600,4252023300096,
%T 44656994071935,401490886656000,3176440229784420,22567393309593600,
%U 146211911499519294,874313719685775360,4872010111798142520,25497827389410525184
%N Coefficients of the modular function J = j - 744.
%C If n=A003173(k)=3 (mod 4) then j(-exp(-sqrt(n) Pi)) is an integer such that exp(sqrt(n) Pi) is very close to an integer, cf. A069014, A056581 and references therein. - M. F. Hasler, Apr 15 2008
%D H. Cohen, Course in Computational Number Theory, page 379.
%D M. Kaneko, Fourier coefficients of the elliptic modular function j(tau) (in Japanese), Rokko Lectures in Mathematics 10, Dept. Math., Faculty of Science, Kobe University, Rokko, Kobe, Japan, 2001.
%D B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 56.
%H Seiichi Manyama, <a href="/A014708/b014708.txt">Table of n, a(n) for n = -1..10000</a> (terms -1..1000 from N. J. A. Sloane)
%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 Miranda C. N. Cheng, John F. R. Duncan, Jeffrey A. Harvey, <a href="http://arxiv.org/abs/1204.2779">Umbral Moonshine</a>, arXiv:1204.2779 [math.RT], Oct 13 2013. See Eq. 1.1.
%H J. Duncan, M. Mertens, K. Ono, <a href="https://arxiv.org/abs/1709.08867">Pariah moonshine</a>, arXiv:1709.08867 [math.RT], 2017. [From Tom Copeland Dec 24 2017]
%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 I. B. Frenkel et al., <a href="http://www.pnas.org/cgi/reprint/81/10/3256.pdf">A natural representation of the Fischer-Griess Monster with the modular function J as character</a>
%H V. G. Kac, <a href="http://www.pnas.org/content/77/9/5048.abstract">A remark on the Conway-Norton Conjecture about the "Monster" simple group</a>, Proc. Nat. Acad. Sci. USA, vol. 77 no. 9 (1980), 5048-5049.
%H E. Klarreich, <a href="https://www.quantamagazine.org/moonshine-link-discovered-for-pariah-symmetries-20170922/">Moonshine link discovered for pariah symmetries</a>, Quanta Magazine, Sep 2017. [From Tom Copeland Dec 24 2017]
%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 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H J. G. Thompson, <a href="http://blms.oxfordjournals.org/content/11/3/352.extract">Some numerology between the Fischer-Griess Monster and the elliptic modular function</a>, Bull. London Math. Soc., 11 (1979), 352-353.
%H University of Sheffield, Department of Pure Mathematics, <a href="http://www.shef.ac.uk/~puremath/theorems/nearint.html">Is e^(Pi*Sqrt(163)) an integer?</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F McKay-Thompson series of class 1A for the Monster group with a(0) = 0.
%F A007245^3/q - 744.
%F a(n) ~ exp(4*Pi*sqrt(n)) / (sqrt(2)*n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2017
%e T1A = 1/q + 196884*q + 21493760*q^2 + 864299970*q^3 + ...
%t a[ n_] := If[ n < 1, Boole[ n==-1 ], SeriesCoefficient[ 1728 KleinInvariantJ[ Log[x] / (2 Pi I)] + x O[x]^n, {x, 0, n}]] (* _Michael Somos_, Jun 29 2011 *)
%o (PARI) {a(n) = if( n<-1, 0, polcoeff( ellj(x + x^3 * O(x^n)) - 744, n))} /* _Michael Somos_, Feb 02 2012 */
%Y Cf. A000521 (the main entry for the j-function), A007240, A027653, A003173, A069014.
%K easy,nonn,nice
%O -1,3
%A _N. J. A. Sloane_
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