A101922 - OEIS

%I #28 Sep 11 2022 09:29:01

%S -1,-1,11,491,11159,-460681,-103577629,-8160790429,624333860399,

%T 386787409545839,68810049201689531,-6999828208693648549,

%U -9872674440874152431161,-3255253386897615662908441,346248578699462435167833491,1072454627614122049417452882131

%N Numerators of expansion of e.g.f. 2^(-1/2) * arcsinh(cos(x)), even powers only.

%C With sign pattern +--+--: numerators of expansion of 2^(-1/2) * arcsinh(cosh(x)).

%C Odd coefficients are zero, denominators are 2^n.

%F arcsinh(cos(x)) = log(sqrt(2)+1) + 1/sqrt(2) * (-(1/2)*x^2/2! - (1/4)*x^4/4! + (11/8)*x^6/6! + (491/16)*x^8/8! + ...).

%F arccosh(cos(x)) = Pi/2 - log(sqrt(2)+1) + 1/sqrt(2) * ((1/2)*x^2/2! + (1/4)*x^4/4! - (11/8)*x^6/6! - (491/16)*x^8/8! - ...).

%F a(n) = - A263246(n). - _Michel Marcus_, Sep 11 2022

%t Table[Numerator[(2n)!SeriesCoefficient[ArcSinh[Cos[x]]/Sqrt[2],{x,0,2n}]],{n,14}] (* _Stefano Spezia_, Aug 29 2022 *)

%o (PARI) lista(nn) = my(x='x + O('x^(nn+1)), p=serlaplace(asinh(cos(x))/sqrt(2))); vector(nn\2, k, round(polcoef(p, 2*k)*2^k)); \\ _Michel Marcus_, Sep 11 2022

%Y Cf. A101924, A012495, A263246.

%K sign

%O 1,3

%A _Ralf Stephan_, Dec 27 2004

%E More terms from _Michel Marcus_, Sep 11 2022