OFFSET
1,2
COMMENTS
Odd coefficients are zero, denominators are 2^n.
FORMULA
arccsch(cos(x)) = log(sqrt(2)+1) + (1/sqrt(2)) * ((1/2)*x^2/2! + (7/4)*x^4/4! + (109/8)*x^6/6! + (3163/16)*x^8/8! + ...).
arcsech(cos(x)) = Pi/2 - log(sqrt(2)+1) - (1/sqrt(2)) * (-(1/2)*x^2/2! + (7/4)*x^4/4! + (109/8)*x^6/6! + (3163/16)*x^8/8! + ...). [warning: this formula appears to be incorrect since arcsech(cos(0)) = 0; - Michel Marcus, Sep 23 2022]
MATHEMATICA
Table[Numerator[(2n)!SeriesCoefficient[ArcCsch[Cos[x]]/Sqrt[2], {x, 0, 2n}]], {n, 14}] (* Stefano Spezia, Aug 29 2022 *)
PROG
(PARI) arccsch(x) = log((1+sqrt(x^2+1))/x);
lista(nn) = localprec(4*nn); my(x='x+O('x^(nn+1)), v=Vec((serlaplace(arccsch(cos(x))))/quadgen(8))); apply(round, vector(#v\2-1, k, v[2*k+1]*2^k)); \\ Michel Marcus, Sep 21 2022
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Ralf Stephan, Dec 27 2004
EXTENSIONS
More terms from Michel Marcus, Sep 20 2022
STATUS
approved