A092924 Expansion of a Schwarzian ({f_{32|8}, tau} / (4*Pi)^2) in powers of q^8.

1, -1008, 8304, -28224, 66672, -127008, 232512, -346752, 533616, -763056, 1046304, -1342656, 1866816, -2215584, 2856576, -3556224, 4269168, -4953312, 6286128, -6914880, 8400672, -9709056, 11060928, -12265344, 14941248, -15877008, 18252192, -20603520, 22935168, -24585120

Offset

0,2

Comments

The q-series f_{32|8} is the g.f. for A082303. This is given on page 274 of McKay and Sebbar along with equation (8.1) which gives an expression for the g.f. A(q) of this sequence. - Michael Somos, Aug 15 2014

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 (21 * E_4(-q) - 16 * E_4(q^2)) / 5 in powers of q. [McKay and Sebbar, equation (8.1)] - Michael Somos, Aug 15 2014

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 16 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 15 2014

Example

G.f. = 1 - 1008*x + 8304*x^2 - 28224*x^3 + 66672*x^4 - 127008*x^5 + 232512*x^6 + ...
G.f. = 1 - 1008*q^8 + 8304*q^16 - 28224*q^24 + 66672*q^32 - 127008*q^40 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; E4[0] := 1; E4[q_]:= 1 +240*Sum[k^3* q^k/(1 - q^k), {k, 1, 150}]; CoefficientList[Series[(21*E4[-q] - 16*E4[q^2])/5, {q, 0, 100}], q] (* G. C. Greubel, Jul 25 2018 *)

Prog

(Sage) A = ModularForms( Gamma0(8), 4, prec=32) . basis(); A[1] - 1008*A[2] + 8304*A[3] + 66672*A[4]; # Michael Somos, Aug 15 2014

Crossrefs

Cf. A062248, A082303.

Sequence in context: A241932 A160451 A254973 * A331770 A290286 A187863

Adjacent sequences: A092921 A092922 A092923 * A092925 A092926 A092927

Keyword

sign

Author

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

Extensions

More terms from Michael Somos, Aug 15 2014

Status

approved