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A007240 McKay-Thompson series of class 1A for the Monster group with a(0) = 24.
(Formerly M5179)
205
1, 24, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184 (list; graph; refs; listen; history; text; internal format)
OFFSET
-1,2
COMMENTS
Changing the term 24 to 744 gives the classical j-function: see A000521 for much more information.
"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]
REFERENCES
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
H. Cohen, Course in Computational Number Theory, page 379.
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.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 56.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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 ].
LINKS
Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..1000 from N. J. A. Sloane)
D. Alexander, C. Cummins, J. McKay and C. Simons, Completely replicable functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, arXiv:math/0611300 [math.NT], 2006-2007.
R. E. Borcherds, Review of "Moonshine Beyond the Monster ..." (Cambridge, 2006), Bull. Amer. Math. Soc., 45 (2008), 675-679.
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).
B. H. Lian and J. L. Wiczer, Genus Zero Modular Functions, arXiv:math/0611291 [math.NT], 2006.
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.
J. G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc., 11 (1979), 352-353.
FORMULA
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
a(n) ~ exp(4*Pi*sqrt(n)) / (sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Jun 28 2018
EXAMPLE
G.f. = 1/q + 24 + 196884*q + 21493760*q^2 + 864299970*q^3 + 20245856256*q^4 + ...
MATHEMATICA
Join[{1, 24}, List @@ Expand[ Normal[ Series[ 1728 * KleinInvariantJ[tau], {tau, 0, 29}]]] /. tau -> 1] // Delete[{{3}, {5}}] (* Jean-François Alcover, Sep 27 2015 *)
PROG
(PARI) {a(n) = if( n<-1, 0, polcoeff( ellj(x + x^3 * O(x^n)) - 720, n))}; /* Michael Somos, May 05 2012 */
(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 */
(PARI) q='q+O('q^66); Vec(ellj(q)-720) \\ Joerg Arndt, Apr 24 2016
CROSSREFS
Sequence in context: A305757 A058550 A145200 * A289029 A287964 A173172
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)