-1,2

Essentially same as McKay-Thompson series of class 1A for Monster.

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.

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 ].

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

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

John Cremona, Home page

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

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

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);

(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).

Sequence in context: A105391 A044984 A119595 this_sequence A066395 A091406 A066396

Adjacent sequences: A000518 A000519 A000520 this_sequence A000522 A000523 A000524

easy,nonn,nice,core

njas