A302609 a(n) = n! * [x^n] exp(n*x)*arctanh(x).

0, 1, 4, 29, 288, 3649, 56160, 1017029, 21181440, 498682881, 13095232000, 379443829709, 12025239367680, 413761766695809, 15360425115176960, 611958601019294325, 26042588632355176448, 1179009749826940037889, 56579126414696034729984, 2868848293506101088635389

Offset: 0

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..384
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A010050, A291484, A293193, A302583, A302584, A302585, A302586, A302587, A302605, A302606, A302608.

Programs

Mathematica

Table[n! SeriesCoefficient[Exp[n x] ArcTanh[x], {x, 0, n}], {n, 0, 19}]
nmax = 20; CoefficientList[Series[Log[(1 - LambertW[-x])/(1 + LambertW[-x])] / (2*(1 + LambertW[-x])), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 09 2019 *)

Formula

E.g.f.: log((1 - LambertW(-x))/(1 + LambertW(-x))) / (2*(1 + LambertW(-x))). - Vaclav Kotesovec, Jun 09 2019

a(n) ~ log(n) * n^n / 4 * (1 + (gamma + 3*log(2))/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 09 2019

a(n) = Sum_{k=1..n} binomial(n,k)*(k-1)!*n^(n-k)*(1-(-1)^k)/2. - Fabian Pereyra, Oct 05 2024