A289638 Coefficients in expansion of -q*E'_8/E_8 where E_8 is the Eisenstein Series (A008410).

-480, 106560, -24577920, 5671616640, -1308807662400, 302026457514240, -69697011105795840, 16083602074756972800, -3711525811469352966240, 856488725919603559612800, -197647268236827050188805760, 45609990487075191657212674560

Offset: 1

Author

Seiichi Manyama, Jul 09 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..422

Crossrefs

-q*E'_k/E_k: A289635 (k=2), A289636 (k=4), A289637 (k=6), this sequence (k=8), A289639 (k=10), A289640 (k=14).

Cf. A006352 (E_2), A008410 (E_8), A287933, A288471.

Programs

Mathematica

nmax = 20; Rest[CoefficientList[Series[-480*x*Sum[k*DivisorSigma[7, k]*x^(k-1), {k, 1, nmax}]/(1 + 480*Sum[DivisorSigma[7, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

a(n) = Sum_{d|n} d * A288471(d).

a(n) = 2*A288261(n)/3 + 16*A000203(n).

a(n) = -Sum_{k=1..n-1} A008410(k)*a(n-k) - A008410(n)*n.

G.f.: 2/3 * E_6/E_4 - 2/3 * E_2 = 2/3 * E_10/E_8 - 2/3 * E_2.

a(n) ~ 2 * (-1)^n * exp(Pi*sqrt(3)*n). - Vaclav Kotesovec, Jul 09 2017