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A289635 Coefficients in expansion of -q*E'_2/E_2 where E_2 is the Eisenstein Series (A006352).

%I A289635 #25 Jul 11 2017 08:47:02
%S A289635 24,720,19296,517920,13893264,372707136,9998360256,268219317312,
%T A289635 7195339794744,193024557070560,5178140391612960,138910500937231488,
%U A289635 3726458885094926160,99967214347459657344,2681753442755678231616
%N A289635 Coefficients in expansion of -q*E'_2/E_2 where E_2 is the Eisenstein Series (A006352).
%H A289635 Seiichi Manyama, <a href="/A289635/b289635.txt">Table of n, a(n) for n = 1..700</a>
%F A289635 a(n) = Sum_{d|n} d * A288968(d).
%F A289635 a(n) = A288877(n)/12 + 2*A000203(n).
%F A289635 a(n) = -Sum_{k=1..n-1} A006352(k)*a(n-k) - A006352(n)*n.
%F A289635 G.f.: 1/12 * E_4/E_2 - 1/12 * E_2.
%F A289635 a(n) ~ 1 / r^n, where r = A211342 = 0.037276810296451658150980785651644618... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24. - _Vaclav Kotesovec_, Jul 09 2017
%e A289635 a(1) = - A006352(1)*1 = 24,
%e A289635 a(2) = -(A006352(1)*a(1)) - A006352(2)*2 = 720,
%e A289635 a(3) = -(A006352(1)*a(2)  + A006352(2)*a(1)) - A006352(3)*3 = 19296,
%e A289635 a(4) = -(A006352(1)*a(3)  + A006352(2)*a(2)  + A006352(3)*a(1)) - A006352(4)*4 = 517920.
%t A289635 nmax = 20; Rest[CoefficientList[Series[24*x*Sum[k*DivisorSigma[1, k]*x^(k-1), {k, 1, nmax}]/(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jul 09 2017 *)
%Y A289635 -q*E'_k/E_k: this sequence (k=2), A289636 (k=4), A289637 (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).
%Y A289635 Cf. A000203, A006352 (E_2), A076835, A211342, A288816, A288877, A288968.
%K A289635 nonn
%O A289635 1,1
%A A289635 _Seiichi Manyama_, Jul 09 2017