A101924 Numerators of expansion of e.g.f. 2^(-1/2) * arccsch(cos(x)), even powers only.
1, 7, 109, 3163, 166201, 14952367, 2002052389, 353291166403, 77829008955121, 21170264082173527, 7106489649576530269, 2913186117837522604843, 1426879448953133350342441, 816516326741659045770111487, 537701607855913139967684905749, 404270165862091267387117902574483
Offset: 1
Programs
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Mathematica
Table[Numerator[(2n)!SeriesCoefficient[ArcCsch[Cos[x]]/Sqrt[2], {x,0,2n}]],{n,14}] (* Stefano Spezia, Aug 29 2022 *) -
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
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]
Extensions
More terms from Michel Marcus, Sep 20 2022
Comments