|
%I M5179 #70 Mar 12 2021 22:24:41
%S 1,24,196884,21493760,864299970,20245856256,333202640600,
%T 4252023300096,44656994071935,401490886656000,3176440229784420,
%U 22567393309593600,146211911499519294,874313719685775360,4872010111798142520,25497827389410525184
%N McKay-Thompson series of class 1A for the Monster group with a(0) = 24.
%C Changing the term 24 to 744 gives the classical j-function: see A000521 for much more information.
%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]
%D Alexander, D.; Cummins, C.; McKay, J.; and Simons, C.; Completely replicable functions, in Groups, Combinatorics & Geometry, (Durham, 1990), pp. 87--98, London Math. Soc. Monograph No. 165. - _N. J. A. Sloane_, Jul 22 2012
%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.
%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).
%D 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 Seiichi Manyama, <a href="/A007240/b007240.txt">Table of n, a(n) for n = -1..10000</a> (terms -1..1000 from N. J. A. Sloane)
%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 A. Berkovich and H. Yesilyurt, <a href="https://arxiv.org/abs/math/0611300">Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms</a>, arXiv:math/0611300 [math.NT], 2006-2007.
%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. 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 D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Comm. Algebra 22, No. 13, 5175-5193 (1994).
%H B. H. Lian and J. L. Wiczer, <a href="https://arxiv.org/abs/math/0611291">Genus Zero Modular Functions</a>, arXiv:math/0611291 [math.NT], 2006.
%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: primorial - 1</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha135.htm">Elliptic modular function j(tau), n=-1 to 100</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha136.htm">n=101 to 200</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha137.htm">n=201 to 300</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha139.htm">n=401 to 500</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha140.htm">n=501 to 600</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha141.htm">n=601 to 700</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha142.htm">n=701 to 800</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha143.htm">n=801 to 900</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha144.htm">n=901 to 1000</a>.
%H Michael Somos, <a href="/A007191/a007191.pdf">Emails to N. J. A. Sloane, 1993</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 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of j(q) - 720 = Theta_Leech(q) / eta(q)^24 in powers of q. Convolution quotient of A008408 and A007240. - _Michael Somos_, May 05 2012
%F a(n) ~ exp(4*Pi*sqrt(n)) / (sqrt(2) * n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2018
%e G.f. = 1/q + 24 + 196884*q + 21493760*q^2 + 864299970*q^3 + 20245856256*q^4 + ...
%t Join[{1, 24}, List @@ Expand[ Normal[ Series[ 1728 * KleinInvariantJ[tau], {tau, 0, 29}]]] /. tau -> 1] // Delete[{{3}, {5}}] (* _Jean-François Alcover_, Sep 27 2015 *)
%o (PARI) {a(n) = if( n<-1, 0, polcoeff( ellj(x + x^3 * O(x^n)) - 720, n))}; /* _Michael Somos_, May 05 2012 */
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = eta(x + x * O(x^n))^24; polcoeff( (1 + 65520 / 691 * (sum( k=1, n, sigma(k, 11) * x^k) - x * A)) / A, n))}; /* _Michael Somos_, May 05 2012 */
%o (PARI) q='q+O('q^66); Vec(ellj(q)-720) \\ _Joerg Arndt_, Apr 24 2016
%Y Cf. A000521, A007240, A008408, A014708.
%K nonn,easy,nice
%O -1,2
%A _N. J. A. Sloane_
|
