A027653 Values of Zagier's function J_1(k) as k runs through the numbers -1, 0, 3, 4, 7, 8, ... which are == -1 or 0 mod 4.

-1, 2, -248, 492, -4119, 7256, -33512, 53008, -192513, 287244, -885480, 1262512, -3493982, 4833456, -12288992, 16576512, -39493539, 52255768, -117966288, 153541020, -331534572, 425691312, -884736744, 1122626864, -2257837845, 2835861520, -5541103056, 6896878512

Offset

1,2

Comments

That is, a(n) = J_1(k) where k is the n-th number >= -1 which is == -1 or 0 mod 4.

References

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.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1002 from N. J. A. Sloane)

D. Zagier, Traces of singular moduli

Formula

For recurrence see Maple code.

a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)). - Vaclav Kotesovec, Oct 06 2019

Maple

with(numtheory); M:=30; t[ -1]:=-1; t[0]:=2;
for n from 1 to M do
t[4*n-1]:=-240*sigma[3](n)-add( r^2*t[4*n-r^2], r=2..floor(sqrt(4*n+1)));
t[4*n]:=-2*add( t[4*n-r^2], r=1..floor(sqrt(4*n+1)));
lprint(t[4*n-1], t[4*n]); od:

Crossrefs

Cf. A027652, A027654, A027655, A014708, A007240, A000521.

Sequence in context: A244926 A024031 A099685 * A024349 A012529 A012527

Adjacent sequences: A027650 A027651 A027652 * A027654 A027655 A027656

Keyword

sign

Author

N. J. A. Sloane

Extensions

Entry revised by N. J. A. Sloane, Jul 24 2006

Status

approved