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%I M5477 N2372 #252 Feb 26 2024 01:27:50
%S 1,744,196884,21493760,864299970,20245856256,333202640600,
%T 4252023300096,44656994071935,401490886656000,3176440229784420,
%U 22567393309593600,146211911499519294,874313719685775360,4872010111798142520,25497827389410525184,126142916465781843075
%N Coefficients of modular function j as power series in q = e^(2 Pi i t). Another name is the elliptic modular invariant J(tau).
%C "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 Changing the term 744 to 24 gives A007240, the McKay-Thompson series of class 1A for Monster simple group.
%C sigma_3(n) is the sum of the cubes of the divisors of n (A001158).
%C Klein's absolute invariant J=j/1728 is Gamma-modular.
%C (n+1)*A000521(n)/24 yields integral values - see A161395. - _Alexander R. Povolotsky_, Jun 09 2009
%C The Mathematica implementation of KleinInvariantJ[] (versions 6 to 8) had bugs giving wrong value for a[7], a[9], a[11] and other values. - _Michael Somos_, Mar 07 2012
%C It is an open question if there are infinitely many k such that a(k) is prime. The known such indices are listed in A339429. See the paper by Fredrik Johansson. - _Peter Luschny_, May 05 2021
%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 115.
%D H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1996, pp. 376ff.
%D A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 20.
%D Evans, David E., and Yasuyuki Kawahigashi. "Subfactors and mathematical physics." Bulletin of the American Mathematical Society, 60:4, (2023), 459-482 (see page 472).
%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 M. J. Knopp, Rademacher on J(tau), Poincare series of nonpositive weights and Eichler cohomology, Notices Amer. Math. Soc., 37:4 (1990), 385-393.
%D S. Lang, Introduction to Modular Forms, Springer-Verlag, 1976, p. 12.
%D B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 56.
%D J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, see p. 482.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A000521/b000521.txt">Table of n, a(n) for n = -1..10000</a> (terms -1..1000 from N. J. A. Sloane)
%H Hans-Fredrick Aas, <a href="http://www.mscand.dk/article/view/10727/8748">Congruences for the Coefficients of the Modular Invariant j(tau)</a>, Mathematica Scandinavica, vol.15, pp. 64-68, 1964.
%H D. Alexander, C. Cummins, J. McKay, and C. Simons, <a href="http://users.rowan.edu/~simons/replicable.pdf">Completely replicable functions</a>, in Groups, Combinatorics & Geometry, (Durham, 1990), pp. 87--98, London Math. Soc. Monograph No. 165.
%H D. Alexander, C. Cummins, J. McKay and C. Simons, <a href="/A007242/a007242_1.pdf">Completely replicable functions</a>, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
%H H. Baier and G. Koehler, <a href="http://projecteuclid.org/euclid.em/1064858788">How to compute the coefficients of the elliptic modular function j(z)</a>, Experimental Mathematics 12 (2003).
%H R. E. Borcherds, <a href="http://dx.doi.org/10.1090/S0273-0979-08-01209-3">Review of "Moonshine Beyond the Monster ..." (Cambridge, 2006)</a>, Bull. Amer. Math. Soc., 45 (2008), 675-679.
%H J. H. Bruinier and K. Ono, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/134.pdf">Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms</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 John Cremona, <a href="http://www.maths.nott.ac.uk/personal/jec">Home page</a>
%H C. Daney, <a href="http://www.openquestions.com/oq-ma017.htm">Open Questions:Elliptic Curves and Modular Forms</a>
%H W. Duke, <a href="http://dx.doi.org/10.1090/S0273-0979-05-01047-5">Continued fractions and modular functions</a>, Bull. Amer. Math. Soc. 42 (2005), 137-162.
%H Andreas Enge, William Hart, and Fredrik Johansson, <a href="http://arxiv.org/abs/1608.06810">Short addition sequences for theta functions</a>, arXiv:1608.06810 [math.NT], 2016-2018.
%H Steven R. Finch, <a href="/A000521/a000521_1.pdf">Modular forms on SL_2(Z)</a>, December 28, 2005. [Cached copy, with permission of the author]
%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 T. Gannon, <a href="http://arxiv.org/abs/math/0109067">Postcards from the edge, or Snapshots of the theory of generalised Moonshine</a>, arXiv:math/0109067 [math.QA], 2001.
%H Y.-H. He and V. Jejjala, <a href="http://arXiv.org/abs/hep-th/0307293">Modular Matrix Models</a>, arXiv:hep-th/0307293, 2003.
%H Yang-Hui He and John McKay, <a href="http://arxiv.org/abs/1408.2083">Moonshine and the Meaning of Life</a>, arXiv:1408.2083 [math.NT], 2014.
%H Yang-Hui He and John McKay, <a href="http://arxiv.org/abs/1505.06742">Sporadic and Exceptional</a>, arXiv:1505.06742 [math.AG], 2015.
%H M. Jankiewicz and T. W. Kephart, <a href="https://doi.org/10.1016/j.nuclphysb.2006.03.030">Transformations among large c conformal field theories</a>, Nucl. Phys. B 744 (2006) 380-397 Table 6.
%H Fredrik Johansson, <a href="https://arxiv.org/abs/2011.14671">Computing isolated coefficients of the j-function</a>, arXiv:2011.14671 [math.NT], 2020.
%H J. Jorgenson, L. Smajlovic, and H. Then, <a href="http://arxiv.org/abs/1309.0648">Kronecker's limit formula, holomorphic modular functions and q-expansions on certain moonshine groups</a>, arXiv preprint arXiv:1309.0648 [math.NT], 2013.
%H M. Kaneko, <a href="https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/60587/1/0965-11.pdf">The Fourier coefficients and the singular moduli of the elliptic modular function j(tau)</a>, Memoirs Faculty Engin. Sci., Kyoto Inst. Technology, 44 (March 1996), pp. 1-5.
%H M. Kaneko and D. Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/ams/stud-adv-math/7/fulltext.pdf">Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials</a>, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998.
%H Masao Koike, <a href="https://oeis.org/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
%H K. Mahler, <a href="http://dx.doi.org/10.1017/S1446788700013367">On a class of non-linear functional equations connected with modular functions</a>, J. Austral. Math. Soc. Ser. A 22 (1976), no. 1, 65--118. MR0441867 (56 #258)
%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/">Factorizations of many number sequences</a>
%H William Stein, <a href="http://wstein.org/">Database</a>
%H Valdo Tatitscheff, <a href="https://arxiv.org/abs/1902.03118">A short introduction to Monstrous Moonshine</a>, arXiv:1902.03118 [math.NT], 2019.
%H J. G. Thompson, <a href="https://doi.org/10.1112/blms/11.3.352">Some numerology between the Fischer-Griess Monster and the elliptic modular function</a>, Bull. London Math. Soc., 11 (1979), 352-353.
%H Jan Vonk, <a href="https://doi.org/10.1090/bull/1700">Overconvergent modular forms and their explicit arithmetic</a>, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/j-Function.html">j-Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MonstrousMoonshine.html">Monstrous Moonshine</a>
%H A. van Wijngaarden, <a href="https://ir.cwi.nl/pub/8906">On the coefficients of the modular invariant J(tau)</a>, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A, 56 (1953), 389-400 [ gives 100 terms ].
%H A. van Wijngaarden, <a href="/A000521/a000521.pdf">On the coefficients of the modular invariant J(tau)</a>, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A, 56 (1953), 389-400 [gives 100 terms]. [Annotated scanned copy]
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Modular_lambda_function">Modular lambda function</a>
%H Herbert S. Zuckerman, <a href="http://dx.doi.org/10.1090/S0002-9904-1939-07116-4">The computation of the smaller coefficients of J(tau)</a>, Bull. Amer. Math. Soc. 45 (1939), 917-919.
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F G.f.: A007245(q)^3/q; or (1 + 240 Sum_{k>0} sigma_3(k) q^k )^3 / (q Product_{k>0} (1-q^k)^24 ).
%F It appears that -n * a(n) = A035230(n). - _Gerald McGarvey_, Dec 21 2006
%F 2 * a(2) = A028520(3). 2 * a(4) + a(1) = A028520(4). 2 * a(6) = A028520(5). - _Gerald McGarvey_, Dec 21 2006
%F 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
%F a(n) ~ exp(4*Pi*n^(1/2))/(2^(1/2)*n^(3/4)) [Petersson (1932), Rademacher (1938)]. - _Gheorghe Coserea_, Oct 09 2015
%F a(n) = (1/n)*(Sum_{r in Z} A027652(n - r^2) + Sum_{r>0, r odd} ((-1)^n * A027652(4*n - r^2) - A027652(16*n - r^2))) for n > 0. - _Seiichi Manyama_, Jun 11 2017
%F a(n) = (1/(n+1))*Sum_{k=1..n+1} (504*A001160(k) - 240*(n-k) * A001158(k)) * a(n-k), a(-1) = 1. - _Seiichi Manyama_, Jul 12 2017
%F G.f.: 256*(1 - lambda + lambda^2)^3/(lambda^2 * (1 - lambda)^2) where lambda is the elliptic modular function (A115977). - _Seiichi Manyama_, Jul 30 2017
%e j = 1/q + 744 + 196884*q + 21493760*q^2 + 864299970*q^3 + 20245856256*q^4 + ...
%e From _Seiichi Manyama_, Jun 11 2017: (Start)
%e a(1) = (1/1)*(A027652(0) + A027652(1) + A027652(0) + (-A027652(3) - A027652(15) - A027652(7))) = (1/1) * 196884 = 196884.
%e a(2) = (1/2)*(A027652(1) + A027652(2) + A027652(1) + (A027652(7) + A027652(-1) - A027652(31) - A027652(23) - A027652(7))) = (1/2) * 42987520 = 21493760.
%e a(3) = (1/3)*(A027652(-1) + A027652(2) + A027652(3) + A027652(2) + A027652(-1) + (-A027652(11) - A027652(3) - A027652(47) - A027652(39) - A027652(23) - A027652(-1))) = (1/3) * 2592899910 = 864299970. (End)
%e If J_n := j(sqrt(-n))^(1/3), then J_1 = 12, J_2 = 20, J_4 = 66, J_77 = 255. - _Michael Somos_, Oct 31 2019
%p with(numtheory): TOP := 31;
%p g2 := (4/3) * (1 + 240 * add(sigma[ 3 ](n)*q^n, n=1..TOP-1));
%p g3 := (8/27) * (1 - 504 * add(sigma[ 5 ](n)*q^n, n=1..TOP-1));
%p delta := series(g2^3 - 27*g3^2, q, TOP);
%p j := series(1728 * g2^3 / delta, q, TOP);
%t CoefficientList[Normal[Series[1728*KleinInvariantJ[z], {z,0,30}]*Exp[ -2*I*Pi/z]] /. E^(Pi*Complex[0,n_]/z) -> t^(-n/2), t] (* _Artur Jasinski_, Dec 20 2008, after Daniel Lichtblau, corrected by _Vaclav Kotesovec_, Jul 07 2020 *)
%t a[ n_] := With[ {tau = Log[q] / (2 Pi I)}, SeriesCoefficient[ Series[ 1728 KleinInvariantJ[ tau], {q, 0, n}], {q, 0, n}]]; (* _Michael Somos_, Nov 20 2011 *)(* Since V7 *)
%t a[ n_] := With[ {e1 = DedekindEta[ Log[q] / (2 Pi I)]^24, e2 = DedekindEta[ Log[q] / (Pi I)]^24}, SeriesCoefficient[ Series[ (e1 + 256 e2)^3 / (e1^2 e2), {q, 0, n + 1}], {q, 0, n}]]; (* _Michael Somos_, Mar 09 2012 *)
%t a[ n_] := With[ {L = ModularLambda[ Log[q] / (2 Pi I)]}, SeriesCoefficient[ Series[ 256 (L^2 - L + 1)^3 / (L (1 - L))^2, {q, 0, 2 n + 3}], {q, 0, n}]]; (* _Michael Somos_, Mar 09 2012 *)
%t a[ n_] := If[ n < -1, 0, With[ {E4 = 1 + 240 Sum[ DivisorSigma[ 3, k] q^k, {k, n + 2}], E6 = 1 - 504 Sum[ DivisorSigma[ 5, k] q^k, {k, n + 2}]}, SeriesCoefficient[ Series[ 1728 E4^3 / (E4^3 - E6^2), {q, 0, n}], {q, 0, n}]]]; (* _Michael Somos_, Mar 09 2012 *)
%t CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^3 / (16777216 * QPochhammer[-1, x]^24), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 23 2017 *)
%t a[n_] := SeriesCoefficient[With[{L = InverseEllipticNomeQ[rootQ]}, 256 (L^2 - L + 1)^3/(L (1 - L))^2], {rootQ, 0, 2n}]; (* _Jan Mangaldan_, Jul 07 2020, after _Michael Somos_; corrected by _Leo C. Stein_, Feb 25 2024 *)
%t a[n_] := SeriesCoefficient[ 12^3 KleinInvariantJ[Log[q]/(2 Pi I)], {q, 0, n}] (* _Leo C. Stein_, Feb 25 2024 *)
%o (PARI) {a(n) = my(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))}; /* _Michael Somos_, Apr 30 2004 */
%o (PARI) {a(n) = my(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 (PARI) {a(n) = my(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 */
%o (PARI) q='q+O('q^66); Vec(ellj(q)) \\ _Joerg Arndt_, Apr 24 2016
%o (PARI) {a(n) = if( n<-1, 0, polcoeff( ellj(x + x^3 * O(x^n)), n))}; /* _Michael Somos_, Dec 25 2016 */
%Y Cf. A005798, A007240, A007245, A014708, A027652, A066395, A078906, A115977, A290403, A290404.
%Y Reversion gives A091406 or A066396.
%Y Cf. A106205 (24th root).
%Y Cf. also A161361, A161362, A161395, A178451, A339429 (indices with prime values).
%K easy,nonn,nice,core
%O -1,2
%A _N. J. A. Sloane_
%E Expanded the definition to include additional search terms. - _N. J. A. Sloane_, Nov 30 2019
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