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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.
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%I #19 Oct 06 2019 05:44:42

%S -1,2,-248,492,-4119,7256,-33512,53008,-192513,287244,-885480,1262512,

%T -3493982,4833456,-12288992,16576512,-39493539,52255768,-117966288,

%U 153541020,-331534572,425691312,-884736744,1122626864,-2257837845,2835861520,-5541103056,6896878512

%N 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.

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

%D 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.

%D 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.

%H Seiichi Manyama, <a href="/A027653/b027653.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1002 from N. J. A. Sloane)

%H D. Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/tex/TracesSingModuli/fulltext.pdf">Traces of singular moduli</a>

%F For recurrence see Maple code.

%F a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)). - _Vaclav Kotesovec_, Oct 06 2019

%p with(numtheory); M:=30; t[ -1]:=-1; t[0]:=2;

%p for n from 1 to M do

%p t[4*n-1]:=-240*sigma[3](n)-add( r^2*t[4*n-r^2],r=2..floor(sqrt(4*n+1)));

%p t[4*n]:=-2*add( t[4*n-r^2],r=1..floor(sqrt(4*n+1)));

%p lprint(t[4*n-1],t[4*n]); od:

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

%K sign

%O 1,2

%A _N. J. A. Sloane_

%E Entry revised by _N. J. A. Sloane_, Jul 24 2006