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%I #47 Jul 11 2021 03:21:32
%S 1,30,-2880,416640,-69178110,12378401280,-2321610157440,
%T 449733567736320,-89200812128140800,18013245273252679710,
%U -3689479088922151082880,764375901202388789804160,-159862757100127037505991680,33699694000689939789618455040,-7152050326608893289997995966720,1526705794390267864554876727856640
%N Coefficients of series whose 8th power is the theta series of E_8 (see A004009).
%D N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
%H Seiichi Manyama, <a href="/A108091/b108091.txt">Table of n, a(n) for n = 0..424</a>
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006.
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://doi.org/10.1016/j.jcta.2006.03.018">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/g4g7.pdf">Seven Staggering Sequences</a>.
%F G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/8). - _Seiichi Manyama_, Jul 02 2017
%F a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(9/8), where c = 3^(1/4) * Gamma(1/3)^(9/4) / (2^(33/8) * Pi^(3/2) * Gamma(7/8)) = 0.1141392450598624077174159151600898926678394937157356242319309115... - _Vaclav Kotesovec_, Jul 02 2017, updated Mar 05 2018
%F a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A300147(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Feb 27 2018
%F G.f.: Sum_{k>=0} A303007(k) * (-f(q))^k where f(q) is Sum_{k>=1} sigma_3(k)*q^k. - _Seiichi Manyama_, Jun 15 2018
%e More precisely, the theta series of E_8 begins 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + ... and the 8th root of this is 1 + 30*q^2 - 2880*q^4 + 416640*q^6 - 69178110*q^8 + ...
%t nmax = 20; s = 8; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/16), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 02 2017 *)
%o (Sage)
%o R.<q> = PowerSeriesRing(ZZ,20)
%o a = R(eisenstein_series_qexp(4,20, normalization='integral'))
%o list(a.sqrt().sqrt().sqrt()) # _Andy Huchala_, Jul 10 2021
%Y E_4^(k/8): this sequence (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5).
%Y Cf. A001158, A004009 (E_4), A106205, A110163, A300147, A303007.
%K sign
%O 0,2
%A _N. J. A. Sloane_ and _Michael Somos_, Jun 06 2005
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