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A062248 Expansion of a Schwarzian ({f_{27|3}, tau} / (4*Pi)^2) in powers of q^3. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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.

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) * 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

EXAMPLE

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 + ...

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-Fran├žois 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 *)

PROG

(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 */

(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 */

CROSSREFS

Cf. A000731, A004009, A062246, A242042.

Sequence in context: A260240 A260062 A235759 * A100146 A235542 A269014

Adjacent sequences:  A062245 A062246 A062247 * A062249 A062250 A062251

KEYWORD

sign

AUTHOR

N. J. A. Sloane, Jul 01 2001

EXTENSIONS

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

More terms from Michael Somos, Aug 12 2014

STATUS

approved

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Last modified January 17 05:26 EST 2019. Contains 319207 sequences. (Running on oeis4.)