A035230 - OEIS

%I #22 Jul 31 2018 14:45:43

%S 1,0,-196884,-42987520,-2592899910,-80983425024,-1666013203000,

%T -25512139800576,-312598958503545,-3211927093248000,

%U -28587962068059780,-225673933095936000,-1608331026494712234,-10491764636229304320,-63336131453375852760,-356969583451747352576

%N Fourier coefficients of T_0.

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

%C T_0 is 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="/A035230/b035230.txt">Table of n, a(n) for n = -1..1000</a>

%F G.f.: G_14/Delta. (in Siegel's notation)

%F 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

%F a(n) = -n*A000521(n). - _Seiichi Manyama_, Jul 12 2017

%F G.f.: -q*j' where j is the elliptic modular invariant (A000521). - _Seiichi Manyama_, Jul 12 2017

%F a(n) ~ -n^(1/4) * exp(4*Pi*sqrt(n)) / sqrt(2). - _Vaclav Kotesovec_, Mar 06 2018

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

%t a[n_] := -n*SeriesCoefficient[1728*KleinInvariantJ[-Log[q]*I/(2*Pi)], {q, 0, n}]; Table[a[n], {n, -1, 14}] (* _Jean-François Alcover_, Nov 02 2017, after _Seiichi Manyama_ *)

%o (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 */

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

%Y Cf. A000521.

%K sign,easy

%O -1,3

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