A062248 Expansion of a Schwarzian ({f_{27|3}, tau} / (4*Pi)^2) in powers of q^3.

1, -48, -216, 1536, -1560, -3024, 13824, -8736, -14040, 41712, -27216, -31968, 112128, -51072, -74304, 193536, -113880, -117936, 375408, -165984, -220752, 528384, -287712, -292032, 898560, -375024, -474768, 1126464, -598848, -585360, 1741824, -722400, -898776

Offset: 0

Author

N. J. A. Sloane, Jul 01 2001

Keywords

• sign

Comments

The q-series f_{27|3} is the g.f. for A062246. This is given on page 274 of McKay and Sebbar along with equation (8.2) which gives an expression for the g.f. A(q) of this sequence, but the left side is A(q^3) and the right side is A(q). - Michael Somos, Aug 12 2014

Ramanujan theta function: f(-q) (see A010815). Ramanujan Lambert series: Q(q) = E_4(q) (see A004009).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 - 48*x - 216*x^2 + 1536*x^3 - 1560*x^4 - 3024*x^5 + 13824*x^6 + ...
G.f. = 1 - 48*q^3 - 216*q^6 + 1536*q^9 - 1560*q^12 - 3024*q^15 + 13824*q^18 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.

Crossrefs

Cf. A000731, A004009, A062246, A242042.

Programs

Magma

A := Basis( ModularForms( Gamma0(9), 8/2), 30); A[1] - 48*A[2] - 216*A[3] + 1536*A[4] - 1560*A[5]; /* Michael Somos, Aug 12 2014 */

Mathematica

QP = QPochhammer; A = x*O[x]^40; A1 = QP[x + A]^3; A3 = QP[x^3 + A]^4; A9 = x*QP[x^9 + A]^3; s = ((A1 + 3*A9)*(A1 + 9*A9)*(A1^2 + 27*A9^2) - 48*x*A3^3 - 216*(A1*A9)^2)/A3; CoefficientList[s, x] (* Jean-François Alcover, Nov 14 2015, adapted from Michael Somos's PARI script *)
eta[q_] := q^(1/24)*QPochhammer[q]; E4[q] := 1; E4[q_] := 1 + 240 *Sum[k^3* q^k/(1 - q^k), {k, 1, 500}]; CoefficientList[Series[E4[q^3] - 48*eta[q^3]^8 - 216*(eta[q]*eta[q^9])^6/eta[q^3]^4, {q, 0, 50}], q] (* G. C. Greubel, May 01 2018 *)

PARI

{a(n) = local(A, A1, A3, A9); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A)^3; A3 = eta(x^3 + A)^4; A9 = x * eta(x^9 + A)^3; polcoeff( ((A1 + 3*A9) * (A1 + 9*A9) * (A1^2 + 27*A9^2) - 48*x*A3^3 - 216*(A1*A9)^2) / A3, n))}; /* Michael Somos, Aug 12 2014 */

Formula

Expansion of Q(q^3) - 48 * q * f(-q^3)^8 - 216 * q^2 * (f(-q) * f(-q)^9)^6 / f(-q^3)^4 in powers of q where Q(), f() are Ramanujan q-series. - Michael Somos, Aug 12 2014

Expansion of (a(q)^4 - 18 * a(q)^3*a(q^3) + 60 * a(q)^2*a(q^3)^2 - 54 * a(q)*a(q^3)^3 + 9 * a(q^3)^4) / -2 where a() is a cubic AGM theta function. - Michael Somos, Aug 12 2014

Expansion of b(q)^4 - 12 * b(q)^3*c(q^3) - 66 * b(q)^2*c(q^3)^2 - 36 * b(q)*c(q^3)^3 + 9 * c(q^3)^4 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Aug 12 2014

Expansion of E_4(q^3) - 48 * eta(q^3)^8 - 216 * eta(q)^6 * eta(q^9)^6 / eta(q^3)^4 in powers of q. [McKay and Sebbar, equation (8.2)] - Michael Somos, Aug 12 2014

G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^4 f(t) where q = exp(2 Pi i t).

a(3*n) = A004009(n) -216 * A242042(3*n). a(3*n + 1) = -48 * A000731(n) -216 * A242042(3*n + 1). a(3*n + 2) = -216 * A242042(3*n + 2). - Michael Somos, Aug 12 2014

Extensions

More terms from John McKay (mckay(AT)cs.concordia.ca), Apr 18 2004

More terms from Michael Somos, Aug 12 2014