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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

Links

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

A296789 Expansion of e.g.f. exp(x*arctanh(x)) (even powers only).

Original entry on oeis.org

1, 2, 20, 504, 24464, 1959840, 234852672, 39370660224, 8799246209280, 2528787321598464, 908585701684024320, 399070678264750356480, 210373049449102957645824, 131083661069772517440921600, 95304505860052894815543705600, 79961055068441273887848131297280
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 20 2017

Keywords

Examples

			exp(x*arctanh(x)) = 1 + 2*x^2/2! + 20*x^4/4! + 504*x^6/6! + 24464*x^8/8! + ...

Crossrefs

Cf. A000246, A009252, A009273, A010050, A166356, A259647, A293193, A296787, A296788.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[Exp[x ArcTanh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 15; Table[(CoefficientList[Series[Exp[x (Log[1 + x] - Log[1 - x])/2], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] exp(x*arctanh(x)).
a(n) ~ 2^(2*n + 2) * n^(2*n) / exp(2*n). - Vaclav Kotesovec, Dec 21 2017
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