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A296680
Expansion of e.g.f. arcsin(arctanh(x)) (odd powers only).
5
1, 3, 53, 2359, 198953, 27412011, 5625656541, 1613676694239, 617477049181521, 304167421243513683, 187546541676182230149, 141512355477854459198343, 128265950128144233675269241, 137512081213377707268891639675, 172108297920263623816775456321325
OFFSET
0,2
LINKS
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}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 18 2017
STATUS
approved