A330495 - OEIS

%I #14 Dec 16 2019 11:40:32

%S 1,6,37,307,2858,32060,405830,5777354,91400200,1593023040,30251766840,

%T 622016655816,13777150847952,327040289212320,8280040187137200,

%U 222696435041359824,6341359225470493440,190609840724078576256,6031297367477133540480,200389374367707186619776

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

%H Vaclav Kotesovec, <a href="/A330495/b330495.txt">Table of n, a(n) for n = 1..400</a>

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

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

%t Table[Sum[(-1)^(n-k) * StirlingS1[n, k] * (k-1)! * DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]

%t 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]!]

%o (PARI) a(n) = sum(k=1, n, (-1)^(n-k)*stirling(n, k, 1)*(k-1)!*sigma(k, 2)); \\ _Michel Marcus_, Dec 16 2019

%Y Cf. A330449, A330450, A330493, A330494.

%K nonn

%O 1,2

%A _Vaclav Kotesovec_, Dec 16 2019