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 A092924 Expansion of a Schwarzian ({f_{32|8}, tau} / (4*Pi)^2) in powers of q^8. 1

%I #6 Jul 26 2018 08:06:22

%S 1,-1008,8304,-28224,66672,-127008,232512,-346752,533616,-763056,

%T 1046304,-1342656,1866816,-2215584,2856576,-3556224,4269168,-4953312,

%U 6286128,-6914880,8400672,-9709056,11060928,-12265344,14941248,-15877008,18252192,-20603520,22935168,-24585120

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

%C 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

%H G. C. Greubel, <a href="/A092924/b092924.txt">Table of n, a(n) for n = 0..1000</a>

%H J. McKay and A. Sebbar, <a href="http://dx.doi.org/10.1007/s002080000116">Fuchsian groups, automorphic functions and Schwarzians</a>, Math. Ann., 318 (2000), 255-275.

%F 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

%F 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

%e G.f. = 1 - 1008*x + 8304*x^2 - 28224*x^3 + 66672*x^4 - 127008*x^5 + 232512*x^6 + ...

%e G.f. = 1 - 1008*q^8 + 8304*q^16 - 28224*q^24 + 66672*q^32 - 127008*q^40 + ...

%t 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 *)

%o (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

%Y Cf. A062248, A082303.

%K sign

%O 0,2

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

%E More terms from _Michael Somos_, Aug 15 2014

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Last modified August 7 11:55 EDT 2024. Contains 375012 sequences. (Running on oeis4.)