OFFSET

0,1

COMMENTS

See A202537 for a guide to related sequences. The Mathematica program includes a graph.

W. Gawronski et al. in their paper - see ref. below - obtained the asymptotics for the Chebyshev-Stirling numbers. In the algebraic description of the respective "asymptotic coefficients" the number x = 2*log phi, where phi is the golden section, play the central role. - Roman Witula, Feb 02 2015

Also two times the Lévy measure for the continued fraction of the golden section, i.e., A202543/log(2) is the mean number of bits gained from the next convergent of the continued fraction representation. (See also Dan Lascu in links.) - A.H.M. Smeets, Jun 06 2018

LINKS

W. Gawronski, L. L. Littlejohn, and T. Neuschel, Asymptotics of Stirling and Chebyshev-Stirling numbers of the second kind, arXiv:1308.6803 [math.CO], 2013.

W. Gawronski, L. L. Littlejohn, and T. Neuschel, Asymptotics of Stirling and Chebyshev-Stirling numbers of the second kind, Studies in Applied Mathematics by MIT 133 (2014), 1-17.

Dan Lascu, A Gauss-Kuzmintype problem for a family of continued fraction expansions, Journal of Number Theory 133 (2013), 2153-2181.

FORMULA

Equals 2*A002390. - A.H.M. Smeets, Jun 06 2018

From Amiram Eldar, Aug 21 2020: (Start)

Equals 2*arcsinh(1/2). [corrected by Georg Fischer, Jul 12 2021]

Equals Sum_{k>=0} (-1)^k*binomial(2*k,k)/((2*k+1)*16^k). (End)

EXAMPLE

0.9624236501192068949955178268487368462703686...

MATHEMATICA

u = 1/2; v = 1/2;

f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1

Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]

r = x /. FindRoot[f[x] == g[x], {x, .9, 1}, WorkingPrecision -> 110]

RealDigits[r] (* A202543 *)

RealDigits[ Log[ (3+Sqrt[5])/2], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)

RealDigits[ FindRoot[ Exp[x/2] == 1 + Exp[-x/2] , {x, 0}, WorkingPrecision -> 128][[1, 2]]][[1]] (* Robert G. Wilson v, Jun 13 2018 *)

PROG

(PARI) 2*asinh(1/2) \\ Michel Marcus, Jun 24 2018, after A002390

CROSSREFS

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Dec 21 2011

EXTENSIONS

Typo in name fixed by Jean-François Alcover, Feb 27 2013

STATUS

approved