A345454 E.g.f.: log(1 - log(1 - x) * exp(x)).

0, 1, 2, 1, -5, 3, 141, 348, -1938, 3013, 274327, 1583338, -4613476, 41135339, 3201505997, 33153080054, 49123558416, 2360520208825, 133442956587099, 2109709010976874, 14751973018988252, 338170133891984663, 15120630911878380457, 324654726628159335686

Offset: 0

Author

Ilya Gutkovskiy, Jul 18 2021

Keywords

• sign

Links

• Index entries for sequences related to logarithmic numbers

Crossrefs

Cf. A002104, A009321, A009324, A191365, A346427.

Programs

Mathematica

nmax = 23; CoefficientList[Series[Log[1 - Log[1 - x] Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
A002104[n_] := A002104[n] = n! Sum[1/((n - k) k!), {k, 0, n - 1}]; a[0] = 0; a[n_] := a[n] = A002104[n] - (1/n) Sum[Binomial[n, k] A002104[n - k] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]

PARI

my(x='x+O('x^25)); concat(0, Vec(serlaplace(log(1 - log(1 - x) * exp(x))))) \\ Michel Marcus, Jul 19 2021

Formula

a(0) = 0; a(n) = A002104(n) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * A002104(n-k) * k * a(k).