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A035230 Fourier coefficients of T_0. 6
1, 0, -196884, -42987520, -2592899910, -80983425024, -1666013203000, -25512139800576, -312598958503545, -3211927093248000, -28587962068059780, -225673933095936000, -1608331026494712234, -10491764636229304320, -63336131453375852760, -356969583451747352576 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,3

COMMENTS

In Siegel's notation, Delta has been normalized already.

T_0 is 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_14/Delta. (in Siegel's notation)

Expansion of (j(q)^4 (j(q) - 1728)^3 Delta(q))^(1/6) in powers of q = exp(2 Pi i t). - Michael Somos, Jul 29 2014

a(n) = -n*A000521(n). - Seiichi Manyama, Jul 12 2017

G.f.: -q*j' where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 12 2017

a(n) ~ -n^(1/4) * exp(4*Pi*sqrt(n)) / sqrt(2). - Vaclav Kotesovec, Mar 06 2018

EXAMPLE

T_0 = 1/q - 196884*q - 42987520*q^2 - 2592899910*q^3 - 80983425024*q^4 + ...

MATHEMATICA

a[n_] := -n*SeriesCoefficient[1728*KleinInvariantJ[-Log[q]*I/(2*Pi)], {q, 0, n}]; Table[a[n], {n, -1, 14}] (* Jean-Fran├žois Alcover, Nov 02 2017, after Seiichi Manyama *)

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, A = x^3 * O(x^n); polcoeff( (x * eta(x + A)^24 * ellj(x + A)^4 * (ellj(x + A) - 1728)^3)^(1/6), n))}; /* Michael Somos, Jul 29 2014 */

(PARI) {a(n) = if( n<-1, 0, polcoeff( -x * ellj(x + x^3 * O(x^n))', n))}; /* Michael Somos, Jul 31 2018 */

CROSSREFS

Cf. A000521.

Sequence in context: A014708 A302407 A305699 * A099818 A185519 A213817

Adjacent sequences:  A035227 A035228 A035229 * A035231 A035232 A035233

KEYWORD

sign,easy

AUTHOR

Barry Brent (barryb(AT)primenet.com)

STATUS

approved

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Last modified May 28 04:00 EDT 2020. Contains 334671 sequences. (Running on oeis4.)