A101922 Numerators of expansion of e.g.f. 2^(-1/2) * arcsinh(cos(x)), even powers only.

-1, -1, 11, 491, 11159, -460681, -103577629, -8160790429, 624333860399, 386787409545839, 68810049201689531, -6999828208693648549, -9872674440874152431161, -3255253386897615662908441, 346248578699462435167833491, 1072454627614122049417452882131

Offset

1,3

Comments

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

Odd coefficients are zero, denominators are 2^n.

Links

Table of n, a(n) for n=1..16.

Formula

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! + ...).

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! - ...).

a(n) = - A263246(n). - Michel Marcus, Sep 11 2022

Mathematica

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

Prog

(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

Crossrefs

Cf. A101924, A012495, A263246.

Sequence in context: A145516 A041931 A322277 * A263246 A263377 A142809

Adjacent sequences: A101919 A101920 A101921 * A101923 A101924 A101925

Keyword

sign

Author

Ralf Stephan, Dec 27 2004

Extensions

More terms from Michel Marcus, Sep 11 2022

Status

approved