A035281 Fourier coefficients of T_4.

1, -240, -141444, -8529280, -238758390, -4303488384, -57655810840, -621029223936, -5646404776905, -44775042057600, -316969789769460, -2038098739288320, -12061935823445274, -66393891923258880, -342773252668461240, -1671250213190122496

Offset

-1,2

Comments

T_4 is the 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_10/Delta (in Siegel's notation.)

G.f.: E_4 * E_6 / Delta where Delta = eta(q)^24. - Michael Somos, Mar 02 2018

a(n) ~ -exp(4*Pi*sqrt(n)) / (sqrt(2) * n^(7/4)). - Vaclav Kotesovec, Oct 04 2020

Example

T_4 = 1/q - 240 - 141444*q - 8529280*q^2 - 238758390*q^3 - 4303488384*q^4 - ...

Mathematica

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 *)
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 *)

Crossrefs

T_k: A035230 (k=0), this sequence (k=4), A035293 (k=6), A035314 (k=8), A035315 (k=10).

Cf. A000594, A004009, A013973.

Sequence in context: A270054 A213696 A275459 * A008696 A047806 A184888

Adjacent sequences: A035278 A035279 A035280 * A035282 A035283 A035284

Keyword

sign,easy,nice

Author

Barry Brent (barryb(AT)primenet.com)

Status

approved