OFFSET
-1,3
COMMENTS
In Siegel's notation, Delta has been normalized already.
T_0 is unique weight-2 normalized meromorphic modular form for SL(2,Z) with all poles at infinity.
REFERENCES
C. L. Siegel, Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bombay, 1980, pp. 249-268.
LINKS
Seiichi Manyama, Table of n, a(n) for n = -1..1000
FORMULA
G.f.: G_14/Delta. (in Siegel's notation)
Expansion of (j(q)^4 (j(q) - 1728)^3 Delta(q))^(1/6) in powers of q = exp(2 Pi i t). - Michael Somos, Jul 29 2014
a(n) = -n*A000521(n). - Seiichi Manyama, Jul 12 2017
G.f.: -q*j' where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 12 2017
a(n) ~ -n^(1/4) * exp(4*Pi*sqrt(n)) / sqrt(2). - Vaclav Kotesovec, Mar 06 2018
EXAMPLE
T_0 = 1/q - 196884*q - 42987520*q^2 - 2592899910*q^3 - 80983425024*q^4 + ...
MATHEMATICA
a[n_] := -n*SeriesCoefficient[1728*KleinInvariantJ[-Log[q]*I/(2*Pi)], {q, 0, n}]; Table[a[n], {n, -1, 14}] (* Jean-François Alcover, Nov 02 2017, after Seiichi Manyama *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, A = x^3 * O(x^n); polcoeff( (x * eta(x + A)^24 * ellj(x + A)^4 * (ellj(x + A) - 1728)^3)^(1/6), n))}; /* Michael Somos, Jul 29 2014 */
(PARI) {a(n) = if( n<-1, 0, polcoeff( -x * ellj(x + x^3 * O(x^n))', n))}; /* Michael Somos, Jul 31 2018 */
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Barry Brent (barryb(AT)primenet.com)
STATUS
approved