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A007240 McKay-Thompson series of class 1A for the Monster group with a(0) = 24.
(Formerly M5179)
197
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

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

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.

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

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.

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.

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..10000

A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms

B. H. Lian and J. L. Wiczer, Genus Zero Modular Functions

Hisanori Mishima, Factorizations of many number sequences: primorial - 1, Elliptic modular function j(tau), n=-1 to 100, n=101 to 200, n=201 to 300, n=401 to 500, n=501 to 600, n=601 to 700, n=701 to 800, n=801 to 900, n=901 to 1000.

Index entries for McKay-Thompson series for Monster simple group

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

EXAMPLE

G.f. = 1/q + 24 + 196884*q + 21493760*q^2 + 864299970*q^3 + 20245856256*q^4 + ...

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 */

CROSSREFS

Cf. A000521, A007240, A008408, A014708.

Sequence in context: A048057 A058550 A145200 * A173172 A061526 A159422

Adjacent sequences:  A007237 A007238 A007239 * A007241 A007242 A007243

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 22 20:39 EDT 2014. Contains 248401 sequences.