A331798 E.g.f.: -log(1 - x) / ((1 - x) * (1 + log(1 - x))).

0, 1, 5, 29, 204, 1714, 16862, 190826, 2447512, 35136696, 558727872, 9754239648, 185546362416, 3820734689472, 84687887312688, 2010622152615504, 50908186083448320, 1369376758488222336, 38998680958184088960, 1172297572938013827456, 37092793335394301708544

Offset: 0

Author

Ilya Gutkovskiy, Jan 26 2020

Keywords

• nonn

Crossrefs

Cf. A000254, A001008, A002805, A003713, A007526, A007840, A215916, A331797.

Programs

Mathematica

nmax = 20; CoefficientList[Series[-Log[1 - x]/((1 - x) (1 + Log[1 - x])), {x, 0, nmax}], x] Range[0, nmax]!
A007526[n_] := n! Sum[1/k!, {k, 0, n - 1}]; a[n_] := Sum[Abs[StirlingS1[n, k]] A007526[k], {k, 0, n}]; Table[a[n], {n, 0, 20}]
A007840[n_] := Sum[Abs[StirlingS1[n, k]] k!, {k, 0, n}]; a[n_] := Sum[Binomial[n, k] k! HarmonicNumber[k] A007840[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

a(n) = Sum_{k=0..n} = |Stirling1(n,k)| * A007526(k).

a(n) = Sum_{k=1..n} binomial(n,k) * k! * H(k) * A007840(n-k), where H(k) is the k-th harmonic number.

a(n) ~ n! / (1 - exp(-1))^(n+1). - Vaclav Kotesovec, Jan 26 2020