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

1,1

Links

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

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

Example

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.

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 *)

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.

Sequence in context: A189412 A246192 A246612 * A105187 A062313 A239815

Adjacent sequences: A289632 A289633 A289634 * A289636 A289637 A289638

Keyword

nonn

Author

Seiichi Manyama, Jul 09 2017

Status

approved