-1,2

"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]

Changing the term 744 to 24 gives A007240, the McKay-Thompson series of class 1A for Monster simple group.

sigma_3(n) is the sum of the cubes of the divisors of n (A001158).

Klein's absolute invariant J=j/1728 is Gamma-modular.

(n+1)*A000521(n)/24 yields integral values - see A161395 [From Alexander R. Povolotsky (pevnev(AT)juno.com), Jun 09 2009]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 07 2009: (Start)

Equals convolution square of A161361: (1, 372, 29250, -134120, 54261375,...)

and row sums of triangle A161362. (End)

R. E. Borcherds, Review of "Moonshine Beyond the Monster ..." (Cambridge, 2006), Bull. Amer. Math. Soc., 45 (2008), 675-679.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 115.

H. Cohen, Course in Computational Number Theory, page 379.

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

W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc., 42 (2005), 137-162.

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 20.

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

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.

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.

S. Lang, Introduction to Modular Forms, Springer-Verlag, 1976, p. 12.

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.

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 56.

J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, see p. 482.

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

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 ].

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.

N. J. A. Sloane, Table of n, a(n) for n = -1..1000

S. R. Finch, Modular forms on SL_2(Z)

H. Baier and G. Koehler, How to compute the coefficients of the elliptic modular function j(z)

John Cremona, Home page

C. Daney, Open Questions:Elliptic Curves and Modular Forms

Hisanori Mishima, Factorizations of many number sequences

William Stein, Database

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Monstrous Moonshine

A007245(q)^3/q; or (1 + 240 sum sigma_3(n) q^n )^3 / (q prod (1-q^n)^24 ) (n=1..inf).

It appears that -n * a(n) = A035230(n). - Gerald McGarvey, Dec 21 2006

2 * a(2) = A028520(3). 2 * a(4) + a(1) = A028520(4). 2 * a(6) = A028520(5). - Gerald McGarvey, Dec 21 2006

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

j = 1/q + 744 + 196884q + 21493760q^2 + 864299970q^3 + ...

with(numtheory): TOP := 31; g2 := 4*Pi^4/3 * (1 + 240 * sum(sigma[ 3 ](n)*q^n, n=1..TOP-1));

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);

j := q -> convert(series(1728 * g2^3 / delta, q, TOP), polynom); jj := j(q);

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]

(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))

(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 */

(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 */

Cf. A014708, A007240, A007245, A066395, A005798, A078906. Reversion gives A091406.

Cf. A106205 (24th root).

Cf. A161361, A161362, A161395.

Adjacent sequences: A000518 A000519 A000520 this_sequence A000522 A000523 A000524

Sequence in context: A105391 A044984 A119595 this_sequence A066395 A161557 A091406

easy,nonn,nice,core,new

N. J. A. Sloane (njas(AT)research.att.com).