|
%I #42 Mar 08 2024 01:15:22
%S 1,1,3,12,65,440,3571,33824,366113,4458240,60321091,897774592,
%T 14576528801,256391130112,4856647308787,98567413125120,
%U 2133825372539585,49080991762153472,1195339768057071619,30729146849826701312,831545527540481198465,23627123985544955559936
%N Expansion of e.g.f.: 1/(cos(x) - x).
%C Radius of convergence |x| < r, where r = cos(r) = 0.739085133215160... (A003957).
%C The continued fraction converges in the whole complex plane, cut along |z|=infinity.
%H G. C. Greubel, <a href="/A200309/b200309.txt">Table of n, a(n) for n = 0..427</a>
%H S. N. Gladkovskii, <a href="http://www.archive.org/details/AnalysisOfTheContinuedFractions.2.3-ed.2009-GladkovskiiSergeiNikolaevich">Analysis Of The Continued Fractions</a> (in Russian).
%F E.g.f.: E(x)=1/(cos(x) - x) = (1-x^2/((x-1)*G(0) + x^3))/(1-x); G(k)= 2*(2*k+1)*(k+1) - x^2 + 2*x^2*(2*k+1)*(k+1)/G(k+1); (continued fraction Euler's kind, 1-step). - _Sergei N. Gladkovskii_, Jan 08 2012
%F E.g.f.: 1/(G(0) - x) where G(k) = 1 - x^2/((4*k+1)*(4*k+2) - x^2*(4*k+1)*(4*k+2)/(x^2 - 4*(k+1)*(4*k+3)/G(k+1) )); - _Sergei N. Gladkovskii_, Dec 16 2012
%F a(n) ~ n!/((sin(r)+1)*r^(n+1)), where r = 0.73908513321516... is the root of the equation r = cos(r). - _Vaclav Kotesovec_, Jun 27 2013
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 12*x^3/3! + 65*x^4/4! + 440*x^5/5! +...
%e where 1/A(x) = 1 - x - x^2/2! + x^4/4! - x^6/6! +...
%e Special values.
%e A(Pi/5) = 10/(5*(sqrt(5)+1)/2 - 2*Pi) = 5.534081362740...
%e A(Pi/6) = 6/(3*sqrt(3) - Pi) = 2.920333635550...
%e A(Pi/8) = 8/(4*sqrt(2+sqrt(2)) - Pi) = 1.882599403781...
%e A(Pi/10) = 10/(5*sqrt(10+sqrt(20))/2 - Pi) = 1.5701119741529...
%e A(Pi/12) = 12/(6*sqrt(2+sqrt(3)) - Pi) = 1.4201994774470...
%t CoefficientList[Series[1/(Cos[x]-x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 27 2013 *)
%o (PARI) {a(n)=n!*polcoeff(1/(cos(x+x*O(x^n))-x),n)}
%o (PARI) x='x+O('x^30); Vec(serlaplace(1/(cos(x) - x))) \\ _G. C. Greubel_, Jul 10 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(Cos(x) - x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jul 10 2018
%Y Cf. A006154, A003957.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 15 2011
| 