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A000521 Coefficients of modular function j as power series in q = e^(2 Pi i t).
(Formerly M5477 N2372)
226
1, 744, 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

"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, Jun 09 2009]

Contribution from Gary W. Adamson, Jun 07 2009: Equals convolution square of A161361: (1, 372, 29250, -134120, 54261375,...) and row sums of triangle A161362.

This sequence is mentioned in the Bruinier-Ono paper in which also appears the finite algebraic formula for the partition function. See chapter 4.1: Singular moduli for j(z). - Omar E. Pol, Nov 03 2011

The Mathematica implementation of KleinInvariantJ[] (versions 6 to 8) has bugs giving wrong value for a[7], a[9], a[11] and other values. - Michael Somos, Mar 07 2012

REFERENCES

Hans-Fredrick Aas: Congruences for the Coefficients of the Modular Invariant j(tau), Mathematica Scandinavica, vol.15, pp. 64-68, 1964. URL: http://www.mscand.dk/issue.php?year=1964&volume=15

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

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, A Course in Computational Algebraic Number Theory, Springer, 1996, pp. 376ff.

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.

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.

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

Mahler, K. On a class of non-linear functional equations connected with modular functions. J. Austral. Math. Soc. Ser. A 22 (1976), no. 1, 65--118. MR0441867 (56 #258)

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.

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).

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 ].

LINKS

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), Experimental Mathematics 12 (2003)

J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms

John Cremona, Home page

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

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

T. Gannon, Postcards from the edge, or Snapshots of the theory of generalised Moonshine, arXiv:math/0109067.

Y.-H. He and V. Jejjala, Modular Matrix Models

J. Jorgenson, L. Smajlovic, H. Then, Kronecker's limit formula, holomorphic modular functions and q-expansions on certain moonshine groups, arXiv preprint arXiv:1309.0648, 2013

Hisanori Mishima, Factorizations of many number sequences

William Stein, Database

Eric Weisstein's World of Mathematics, j-Function

Eric Weisstein's World of Mathematics, Monstrous Moonshine

Index entries for McKay-Thompson series for Monster simple group

FORMULA

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

EXAMPLE

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

MAPLE

with(numtheory): TOP := 31;

g2 := (4/3) * (1 + 240 * add(sigma[ 3 ](n)*q^n, n=1..TOP-1));

g3 := (8/27) * (1 - 504 * add(sigma[ 5 ](n)*q^n, n=1..TOP-1));

delta := series(g2^3 - 27*g3^2, q, TOP);

j := series(1728 * g2^3 / delta, q, TOP);

MATHEMATICA

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, Dec 20 2008]

a[ n_] := With[ {tau = Log[q] / (2 Pi I)}, SeriesCoefficient[ Series[ 1728 KleinInvariantJ[ tau], {q, 0, n}], {q, 0, n}]]; (* Michael Somos, Nov 20 2011 *)(* Since V7 *)

a[ n_] := With[ {e1 = DedekindEta[ Log[q] / (2 Pi I)]^24, e2 = DedekindEta[ Log[q] / (Pi I)]^24}, SeriesCoefficient[ Series[ (e1 + 256 e2)^3 / (e1^2 e2), {q, 0, n + 1}], {q, 0, n}]]; (* Michael Somos, Mar 09 2012 *)

a[ n_] := With[ {L = ModularLambda[ Log[q] / (2 Pi I)]}, SeriesCoefficient[ Series[ 256 (L^2 - L + 1)^3 / (L (1 - L))^2, {q, 0, 2 n + 3}], {q, 0, n}]]; (* Michael Somos, Mar 09 2012 *)

a[ n_] := If[ n < -1, 0, With[ {E4 = 1 + 240 Sum[ DivisorSigma[ 3, k] q^k, {k, n + 2}], E6 = 1 - 504 Sum[ DivisorSigma[ 5, k] q^k, {k, n + 2}]}, SeriesCoefficient[ Series[ 1728 E4^3 / (E4^3 - E6^2), {q, 0, n}], {q, 0, n}]]]; (* Michael Somos, Mar 09 2012 *)

PROG

(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 */

CROSSREFS

Cf. A014708, A007240, A007245, A066395, A005798, A078906. Reversion gives A091406 or A066396.

Cf. A106205 (24th root).

Cf. also A161361, A161362, A161395, A178451.

Sequence in context: A235252 A210178 A192731 * A178449 A178451 A066395

Adjacent sequences:  A000518 A000519 A000520 * A000522 A000523 A000524

KEYWORD

easy,nonn,nice,core

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 25 04:22 EDT 2014. Contains 240994 sequences.