|
|
A014708
|
|
Coefficients of the modular function J = j - 744.
|
|
214
|
|
|
1, 0, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
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]
|
|
FORMULA
|
McKay-Thompson series of class 1A for the Monster group with a(0) = 0.
|
|
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
|
|
|
KEYWORD
|
easy,nonn,nice
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|