%I A000521 M5477 N2372
%S A000521 1,744,196884,21493760,864299970,20245856256,333202640600,4252023300096,
%T A000521 44656994071935,401490886656000,3176440229784420,22567393309593600,
%U A000521 146211911499519294,874313719685775360,4872010111798142520,25497827389410525184
%N A000521 Coefficients of modular function j as power series in q=e^(2Pi i t).
%C A000521 "The most natural normalization [of the j function] is to set the constant 
               term equal to 24, the number given by Rademacher's infinite series 
               for coefficients of the j function". [Borcherds]
%C A000521 Changing the term 744 to 24 gives A007240, the McKay-Thompson series 
               of class 1A for Monster simple group.
%C A000521 sigma_3(n) is the sum of the cubes of the divisors of n (A001158).
%C A000521 Klein's absolute invariant J=j/1728 is Gamma-modular.
%C A000521 (n+1)*A000521(n)/24 yields integral values - see A161395 [From Alexander 
               R. Povolotsky (pevnev(AT)juno.com), Jun 09 2009]
%C A000521 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 07 2009: 
               (Start)
%C A000521 Equals convolution square of A161361: (1, 372, 29250, -134120, 54261375,
               ...)
%C A000521 and row sums of triangle A161362. (End)
%D A000521 R. E. Borcherds, Review of "Moonshine Beyond the Monster ..." (Cambridge, 
               2006), Bull. Amer. Math. Soc., 45 (2008), 675-679.
%D A000521 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 115.
%D A000521 H. Cohen, Course in Computational Number Theory, page 379.
%D A000521 J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. 
               Soc. 11 (1979) 308-339.
%D A000521 W. Duke, Continued fractions and modular functions, Bull. Amer. Math. 
               Soc., 42 (2005), 137-162.
%D A000521 A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 
               3, p. 20.
%D A000521 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. 
               Algebra 22, No. 13, 5175-5193 (1994).
%D A000521 M. Kaneko, The Fourier coefficients and the singular moduli of the elliptic 
               modular function j(tau), Memoirs Faculty Engin. Sci., Kyoto Inst. 
               Technology, 44 (March 1996), pp. 1-5.
%D A000521 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 A000521 M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series 
               and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and 
               J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, 
               Amer. Math. Soc., 1998.
%D A000521 S. Lang, Introduction to Modular Forms, Springer-Verlag, 1976, p. 12.
%D A000521 J. McKay and H. Strauss, The q-series of monstrous moonshine and the 
               decomposition of the head characters. Comm. Algebra 18 (1990), no. 
               1, 253-278.
%D A000521 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, 
               p. 56.
%D A000521 J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, 
               Springer, see p. 482.
%D A000521 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000521 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000521 J. G. Thompson, Some numerology between the Fischer-Griess Monster and 
               the elliptic modular function, Bull. London Math. Soc., 11 (1979), 
               352-353.
%D A000521 A. van Wijngaarden, On the coefficients of the modular invariant J(tau), 
               Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, 
               Series A, 56 (1953), 389-400 [ gives 100 terms ].
%H A000521 N. J. A. Sloane, <a href="b000521.txt">Table of n, a(n) for n = -1..1000</
               a>
%H A000521 S. R. Finch, <A HREF="http://algo.inria.fr/bsolve/">Modular forms on 
               SL_2(Z)</A>
%H A000521 H. Baier and G. Koehler, <a href="http://www.expmath.org/expmath/volumes/
               12/12.html">How to compute the coefficients of the elliptic modular 
               function j(z)</a>
%H A000521 John Cremona, <a href="http://www.maths.nott.ac.uk/personal/jec">Home 
               page</a>
%H A000521 C. Daney, <a href="http://www.openquestions.com/oq-ma017.htm">Open Questions:Elliptic 
               Curves and Modular Forms</a>
%H A000521 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha103.htm">Factorizations of many number sequences</a>
%H A000521 William Stein, <a href="http://modular.math.washington.edu/">Database</
               a>
%H A000521 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               j-Function.html">Link to a section of The World of Mathematics.</
               a>
%H A000521 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MonstrousMoonshine.html">Monstrous Moonshine</a>
%H A000521 <a href="Sindx_Mat.html#McKay_Thompson">Index entries for McKay-Thompson 
               series for Monster simple group</a>
%F A000521 A007245(q)^3/q; or (1 + 240 sum sigma_3(n) q^n )^3 / (q prod (1-q^n)^24 
               ) (n=1..inf).
%F A000521 It appears that -n * a(n) = A035230(n). - Gerald McGarvey, Dec 21 2006
%F A000521 2 * a(2) = A028520(3). 2 * a(4) + a(1) = A028520(4). 2 * a(6) = A028520(5). 
               - Gerald McGarvey, Dec 21 2006
%F A000521 Expansion of 128 * (theta_2(q)^8 + theta_3(q)^8 + theta_4(q)^8) * (theta_2(q)^-8 
               + theta_3(q)^-8 + theta_4(q)^-8) in powers of q^2. - Michael Somos 
               Oct 02 2007
%e A000521 j = 1/q + 744 + 196884q + 21493760q^2 + 864299970q^3 + ...
%p A000521 with(numtheory): TOP := 31; g2 := 4*Pi^4/3 * (1 + 240 * sum(sigma[ 3 
               ](n)*q^n,n=1..TOP-1));
%p A000521 g3 := 8*Pi^6/27 * (1 - 504 * sum(sigma[ 5 ](n)*q^n,n=1..TOP-1)); delta 
               := convert(series(g2^3 - 27*g3^2, q, TOP), polynom);
%p A000521 j := q -> convert(series(1728 * g2^3 / delta, q, TOP), polynom); jj := 
               j(q);
%t A000521 CoefficientList[Series[1728*KleinInvariantJ[z], {z,0,10}]*Exp[ -2*I*Pi/
               z] /. E^(Pi*Complex[0,n_]/z) -> t^(-n/2), t] (*Daniel Lichtblau*) 
               [From Artur Jasinski (grafix(AT)csl.pl), Dec 20 2008]
%o A000521 (PARI) a(n)=local(A); if(n<-1,0,A=x^(2*n+2)*O(x);A=x*(eta(x+A)*eta(x^4+A)^2/
               eta(x^2+A)^3)^8; polcoeff(subst(256*(1-x+x^2)^3/(x-x^2)^2,x,16*A),
               2*n))
%o A000521 (PARI) a(n)=local(A); if(n<-1,0,A=x^(5*n+5)*O(x);A=(eta(x+A)/eta(x^5+A))^6/
               x; polcoeff(subst( (x^2+10*x+5)^3/x,x,A),5*n)) /* Michael Somos Apr 
               30 2004 */
%o A000521 (PARI) a(n)=local(A); if(n<-1,0,A=x^2*O(x^n); A=x*(eta(x^2+A)/eta(x+A))^24; 
               polcoeff((1+256*A)^3/A,n)) /* Michael Somos Jul 13 2004 */
%Y A000521 Cf. A014708, A007240, A007245, A066395, A005798, A078906. Reversion gives 
               A091406.
%Y A000521 Cf. A106205 (24th root).
%Y A000521 Cf. A161361, A161362, A161395.
%Y A000521 Sequence in context: A105391 A044984 A119595 this_sequence A066395 A161557 
               A091406
%Y A000521 Adjacent sequences: A000518 A000519 A000520 this_sequence A000522 A000523 
               A000524
%K A000521 easy,nonn,nice,core
%O A000521 -1,2
%A A000521 N. J. A. Sloane (njas(AT)research.att.com).