|
%I #35 Mar 05 2018 09:30:53
%S 1,-84,-20412,-6617856,-2505409788,-1027549673640,-442991672331264,
%T -197605206331169280,-90359564898413083644,-42105781947560460595284,
%U -19913609001700051596476280,-9531377528273693889501019392
%N Coefficients in expansion of E_6^(1/6).
%H Seiichi Manyama, <a href="/A289325/b289325.txt">Table of n, a(n) for n = 0..367</a>
%H R. S. Maier, <a href="http://arxiv.org/abs/0807.1081">Nonlinear differential equations satisfied by certain classical modular forms</a>, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.30).
%F G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/6).
%F G.f.: 2F1(1/12, 7/12; 1; 1728/(1728-j)) where j is the elliptic modular invariant (A000521). - _Seiichi Manyama_, Jul 07 2017
%F a(n) ~ c * exp(2*Pi*n) / n^(7/6), where c = -Gamma(1/4)^(8/3) * Gamma(1/3)^2 / (2^(9/2) * 3^(1/6) * Pi^(7/2)) = -0.149083170913265334790743918765758886634155... - _Vaclav Kotesovec_, Jul 08 2017, updated Mar 05 2018
%e From _Seiichi Manyama_, Jul 08 2017: (Start)
%e 2F1(1/12, 7/12; 1; 1728/(1728 - j))
%e = 1 - A289557(1)/(j - 1728) + A289557(2)/(j - 1728)^2 - A289557(3)/(j - 1728)^3 + ...
%e = 1 - 84/(j - 1728) + 62244/(j - 1728)^2 - 64318800/(j - 1728)^3 + ...
%e = 1 - 84*q - 82656*q^2 - 64795248*q^3 - ...
%e + 62244*q^2 + 122496192*q^3 + ...
%e - 64318800*q^3 - ...
%e + ...
%e = 1 - 84*q - 20412*q^2 - 6617856*q^3 - ... (End)
%t nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(1/6), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 08 2017 *)
%Y E_6^(k/12): A109817 (k=1), this sequence (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).
%Y Cf. A013973 (E_6), A288851, A289417, A289557.
%K sign
%O 0,2
%A _Seiichi Manyama_, Jul 02 2017
| 