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%I #16 Mar 16 2024 10:12:34
%S 1,14,-544,34496,-2512254,197053696,-16194254272,1374326128896,
%T -119403428951808,10561444878559246,-947458249960057024,
%U 85971010094510200128,-7874673015172093889024,727016151987267244001536,-67573426491012510177925760,6317185611058637805840976640
%N Expansion of eighth root of theta series of D_8 lattice.
%H Vaclav Kotesovec, <a href="/A109773/b109773.txt">Table of n, a(n) for n = 0..500</a>
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%F a(n) ~ -(-1)^n * c * d^n / n^(9/8), where d = -1/EllipticNomeQ(-3+2*sqrt(2)) = 101.05698591144255836558034070124390358691255555299851256465840129800034600429... and c = 0.11312909975079493828483346745366595624358550348529207605517154972... - _Vaclav Kotesovec_, Dec 11 2017, updated Mar 16 2024
%t terms = 16; f[q_] = LatticeData["D8", "ThetaSeriesFunction"][-I Log[q]/Pi]; s = Series[f[q]^(1/8), {q, 0, 2 terms}]; CoefficientList[s, q^2][[1 ;; terms]] // Round (* _Jean-François Alcover_, Jul 07 2017 *)
%t CoefficientList[Series[((EllipticTheta[3, 0, Sqrt[x]]^8 + EllipticTheta[4, 0, Sqrt[x]]^8)/2)^(1/8), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Dec 10 2017 *)
%Y Cf. A008430, A008427, A109772.
%K sign
%O 0,2
%A _N. J. A. Sloane_ and _Nadia Heninger_, Aug 13 2005
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