A330495 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * (k-1)! * sigma_2(k), where sigma_2 = A001157.

1, 6, 37, 307, 2858, 32060, 405830, 5777354, 91400200, 1593023040, 30251766840, 622016655816, 13777150847952, 327040289212320, 8280040187137200, 222696435041359824, 6341359225470493440, 190609840724078576256, 6031297367477133540480, 200389374367707186619776

Offset: 1

Author

Vaclav Kotesovec, Dec 16 2019

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..400

Crossrefs

Cf. A330449, A330450, A330493, A330494.

Programs

Mathematica

Table[Sum[(-1)^(n-k) * StirlingS1[n, k] * (k-1)! * DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[Log[1/(1 - x)]^k/(k*(1 - Log[1/(1 - x)]^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]

PARI

a(n) = sum(k=1, n, (-1)^(n-k)*stirling(n, k, 1)*(k-1)!*sigma(k, 2)); \\ Michel Marcus, Dec 16 2019

Formula

E.g.f.: Sum_{k>=1} log(1/(1 - x))^k / (k * (1 - log(1/(1 - x))^k)^2).

a(n) ~ n! * zeta(3) * n * exp(n) / (exp(1) - 1)^(n+2).