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Fourier coefficients of T_{10}.
3

%I #18 Oct 04 2020 07:37:58

%S 1,264,8244,139520,1672290,15872256,126745880,884100096,5525046495,

%T 31498809600,166049246340,817866616320,3794952949854,16699329285120,

%U 70071039813240,281650911606784,1088671630120515,4060062852952320

%N Fourier coefficients of T_{10}.

%C T_{10} is the unique weight = -8 normalized meromorphic modular form for SL(2,Z) with all poles at infinity.

%D C. L. Siegel, Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bombay, 1980, pp. 249-268.

%H Seiichi Manyama, <a href="/A035315/b035315.txt">Table of n, a(n) for n = -1..1000</a>

%H Borcherds, Richard E., <a href="http://arXiv.org/abs/alg-geom/9609022">Automorphic forms with singularities on Grassmannians</a>, Invent. Math. 132 (1998), 491-562.

%F G.f.: G_4/Delta (in Siegel's notation.)

%F a(n) ~ exp(4*Pi*sqrt(n)) / (sqrt(2) * n^(19/4)). - _Vaclav Kotesovec_, Oct 04 2020

%e T_{10} = 1/q + 264 + 8244 q + ....

%o (PARI) {a(n)=if(n<-1, 0, n++; polcoeff( sum(k=1,n,240*sigma(k,3)*x^k,1+x*O(x^n))/eta(x+x*O(x^n))^24,n))} /* _Michael Somos_, Apr 12 2005 */

%K easy,nonn

%O -1,2

%A Barry Brent (barryb(AT)primenet.com)