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%I #18 Oct 04 2020 07:38:16
%S 1,-240,-141444,-8529280,-238758390,-4303488384,-57655810840,
%T -621029223936,-5646404776905,-44775042057600,-316969789769460,
%U -2038098739288320,-12061935823445274,-66393891923258880,-342773252668461240,-1671250213190122496
%N Fourier coefficients of T_4.
%C T_4 is the unique weight = -2 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="/A035281/b035281.txt">Table of n, a(n) for n = -1..1000</a>
%F G.f.: G_10/Delta (in Siegel's notation.)
%F G.f.: E_4 * E_6 / Delta where Delta = eta(q)^24. - _Michael Somos_, Mar 02 2018
%F a(n) ~ -exp(4*Pi*sqrt(n)) / (sqrt(2) * n^(7/4)). - _Vaclav Kotesovec_, Oct 04 2020
%e T_4 = 1/q - 240 - 141444*q - 8529280*q^2 - 238758390*q^3 - 4303488384*q^4 - ...
%t a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, (t2^2 + 14 t2 t3 + t3^2) (t2^3 - 33 (t2 + t3) t2 t3 + t3^3) / (q QPochhammer[ q]^24)], {q, 0, n}]; (* _Michael Somos_, Mar 02 2018 *)
%t a[ n_] := SeriesCoefficient[ With[{t3 = EllipticTheta[ 3, 0, q]^4, t4 = EllipticTheta[ 4, 0, q]^4}, (16 t3^2 - 16 t3 t4 + t4^2) (64 t3^3 - 96 t3^2 t4 + 30 t3 t4^2 + t4^3) / (- q QPochhammer[ q]^24)], {q, 0, n}]; (* _Michael Somos_, Mar 02 2018 *)
%Y T_k: A035230 (k=0), this sequence (k=4), A035293 (k=6), A035314 (k=8), A035315 (k=10).
%Y Cf. A000594, A004009, A013973.
%K sign,easy,nice
%O -1,2
%A Barry Brent (barryb(AT)primenet.com)