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

24, 720, 19296, 517920, 13893264, 372707136, 9998360256, 268219317312, 7195339794744, 193024557070560, 5178140391612960, 138910500937231488, 3726458885094926160, 99967214347459657344, 2681753442755678231616

Offset: 1

Author

Seiichi Manyama, Jul 09 2017

Keywords

• nonn

Examples

			a(1) = - A006352(1)*1 = 24,
a(2) = -(A006352(1)*a(1)) - A006352(2)*2 = 720,
a(3) = -(A006352(1)*a(2)  + A006352(2)*a(1)) - A006352(3)*3 = 19296,
a(4) = -(A006352(1)*a(3)  + A006352(2)*a(2)  + A006352(3)*a(1)) - A006352(4)*4 = 517920.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..700

Crossrefs

-q*E'_k/E_k: this sequence (k=2), A289636 (k=4), A289637 (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).

Cf. A000203, A006352 (E_2), A076835, A211342, A288816, A288877, A288968.

Programs

Mathematica

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 *)

Formula

a(n) = Sum_{d|n} d * A288968(d).

a(n) = A288877(n)/12 + 2*A000203(n).

a(n) = -Sum_{k=1..n-1} A006352(k)*a(n-k) - A006352(n)*n.

G.f.: 1/12 * E_4/E_2 - 1/12 * E_2.

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