%I #30 Mar 01 2018 02:43:46

%S -240,53280,-12288960,2835808320,-654403831200,151013228757120,

%T -34848505552897920,8041801037378486400,-1855762905734676483120,

%U 428244362959801779806400,-98823634118413525094402880,22804995243537595828606337280

%N Coefficients in expansion of -q*E'_4/E_4 where E_4 is the Eisenstein Series (A004009).

%H Seiichi Manyama, <a href="/A289636/b289636.txt">Table of n, a(n) for n = 1..422</a>

%F a(n) = Sum_{d|n} d * A110163(d) = A289633(n)/6.

%F a(n) = A288261(n)/3 + 8*A000203(n).

%F a(n) = -Sum_{k=1..n-1} A004009(k)*a(n-k) - A004009(n)*n.

%F G.f.: 1/3 * E_6/E_4 - 1/3 * E_2.

%F a(n) ~ (-1)^n * exp(Pi*sqrt(3)*n). - _Vaclav Kotesovec_, Jul 09 2017

%e a(1) = 1 * A110163(1) = -240,

%e a(2) = 1 * A110163(1) + 2 * A110163(2) = 53280,

%e a(3) = 1 * A110163(1) + 3 * A110163(3) = -12288960.

%t nmax = 20; Rest[CoefficientList[Series[-240*x*Sum[k*DivisorSigma[3, k]*x^(k-1), {k, 1, nmax}]/(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jul 09 2017 *)

%t terms = 12; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[-D[Ei[4], x]/Ei[4] + O[x]^terms, x] (* _Jean-François Alcover_, Mar 01 2018 *)

%Y -q*E'_k/E_k: A289635 (k=2), this sequence (k=4), A289637 (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).

%Y Cf. A001943, A004009 (E_4), A006352 (E_2), A110163, A145094, A289633.

%K sign

%O 1,1

%A _Seiichi Manyama_, Jul 09 2017