A014708 Coefficients of the modular function J = j - 744.

1, 0, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184

Offset

-1,3

Comments

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

References

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.

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..1000 from N. J. A. Sloane)

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

Miranda C. N. Cheng, John F. R. Duncan, Jeffrey A. Harvey, Umbral Moonshine, arXiv:1204.2779 [math.RT], Oct 13 2013. See Eq. 1.1.

J. Duncan, M. Mertens, K. Ono, Pariah moonshine, arXiv:1709.08867 [math.RT], 2017. [From Tom Copeland Dec 24 2017]

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

I. B. Frenkel et al., A natural representation of the Fischer-Griess Monster with the modular function J as character

V. G. Kac, A remark on the Conway-Norton Conjecture about the "Monster" simple group, Proc. Nat. Acad. Sci. USA, vol. 77 no. 9 (1980), 5048-5049.

E. Klarreich, Moonshine link discovered for pariah symmetries, Quanta Magazine, Sep 2017. [From Tom Copeland Dec 24 2017]

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.

Hisanori Mishima, Factorizations of many number sequences

J. G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc., 11 (1979), 352-353.

University of Sheffield, Department of Pure Mathematics, Is e^(Pi*Sqrt(163)) an integer?

Index entries for McKay-Thompson series for Monster simple group

Formula

McKay-Thompson series of class 1A for the Monster group with a(0) = 0.

A007245^3/q - 744.

a(n) ~ exp(4*Pi*sqrt(n)) / (sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Jun 28 2017

Example

T1A = 1/q + 196884*q + 21493760*q^2 + 864299970*q^3 + ...

Mathematica

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 *)

Prog

(PARI) {a(n) = if( n<-1, 0, polcoeff( ellj(x + x^3 * O(x^n)) - 744, n))} /* Michael Somos, Feb 02 2012 */

Crossrefs

Cf. A000521 (the main entry for the j-function), A007240, A027653, A003173, A069014.

Sequence in context: A113919 A001379 A247242 * A302407 A305699 A035230

Adjacent sequences: A014705 A014706 A014707 * A014709 A014710 A014711

Keyword

easy,nonn,nice

Author

N. J. A. Sloane

Status

approved