A296680 Expansion of e.g.f. arcsin(arctanh(x)) (odd powers only).

1, 3, 53, 2359, 198953, 27412011, 5625656541, 1613676694239, 617477049181521, 304167421243513683, 187546541676182230149, 141512355477854459198343, 128265950128144233675269241, 137512081213377707268891639675, 172108297920263623816775456321325

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..215

Formula

E.g.f.: arcsinh(arctan(x)) (odd powers only, absolute values).

E.g.f.: -i*log((i/2)*(log(1 + x) - log(1 - x)) + sqrt(1 - (log(1 + x) - log(1 - x))^2/4)), where i is the imaginary unit (odd powers only).

Example

arcsin(arctanh(x)) = x/1! + 3*x^3/3! + 53*x^5/5! + 2359*x^7/7! + 198953*x^9/9! + 27412011*x^11/11! + ...

Maple

S:= series(arcsin(arctanh(x)), x, 52):
seq(coeff(S, x, n)*n!, n=1..51, 2); # Robert Israel, Dec 18 2017

Mathematica

nmax = 15; Table[(CoefficientList[Series[ArcSin[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 15; Table[(CoefficientList[Series[-I Log[(I/2) (Log[1 + x] - Log[1 - x]) + Sqrt[1 - (Log[1 + x] - Log[1 - x])^2/4]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Crossrefs

Cf. A001818, A003717, A010050, A012134, A296464, A296466, A296679.

Sequence in context: A227051 A300606 A301348 * A118194 A325725 A173357

Adjacent sequences: A296677 A296678 A296679 * A296681 A296682 A296683

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 18 2017

Status

approved