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%I #25 Sep 23 2022 03:00:46
%S 1,7,109,3163,166201,14952367,2002052389,353291166403,77829008955121,
%T 21170264082173527,7106489649576530269,2913186117837522604843,
%U 1426879448953133350342441,816516326741659045770111487,537701607855913139967684905749,404270165862091267387117902574483
%N Numerators of expansion of e.g.f. 2^(-1/2) * arccsch(cos(x)), even powers only.
%C Odd coefficients are zero, denominators are 2^n.
%F 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! + ...).
%F 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]
%t Table[Numerator[(2n)!SeriesCoefficient[ArcCsch[Cos[x]]/Sqrt[2], {x,0,2n}]],{n,14}] (* _Stefano Spezia_, Aug 29 2022 *)
%o (PARI) arccsch(x) = log((1+sqrt(x^2+1))/x);
%o 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
%Y Cf. A101922, A012495.
%K nonn,frac
%O 1,2
%A _Ralf Stephan_, Dec 27 2004
%E More terms from _Michel Marcus_, Sep 20 2022
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