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

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

Offset

0,2

Links

Table of n, a(n) for n=0..15.

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

Example

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

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}]

Crossrefs

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

Sequence in context: A279837 A279200 A277414 * A168136 A103353 A009344

Adjacent sequences: A296786 A296787 A296788 * A296790 A296791 A296792

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 20 2017

Status

approved