A200309 Expansion of e.g.f.: 1/(cos(x) - x).

1, 1, 3, 12, 65, 440, 3571, 33824, 366113, 4458240, 60321091, 897774592, 14576528801, 256391130112, 4856647308787, 98567413125120, 2133825372539585, 49080991762153472, 1195339768057071619, 30729146849826701312, 831545527540481198465, 23627123985544955559936

Offset

0,3

Comments

Radius of convergence |x| < r, where r = cos(r) = 0.739085133215160... (A003957).

The continued fraction converges in the whole complex plane, cut along |z|=infinity.

Links

G. C. Greubel, Table of n, a(n) for n = 0..427

S. N. Gladkovskii, Analysis Of The Continued Fractions (in Russian).

Formula

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

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

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

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 12*x^3/3! + 65*x^4/4! + 440*x^5/5! +...
where 1/A(x) = 1 - x - x^2/2! + x^4/4! - x^6/6! +...
Special values.
A(Pi/5) = 10/(5*(sqrt(5)+1)/2 - 2*Pi) = 5.534081362740...
A(Pi/6) = 6/(3*sqrt(3) - Pi) = 2.920333635550...
A(Pi/8) = 8/(4*sqrt(2+sqrt(2)) - Pi) = 1.882599403781...
A(Pi/10) = 10/(5*sqrt(10+sqrt(20))/2 - Pi) = 1.5701119741529...
A(Pi/12) = 12/(6*sqrt(2+sqrt(3)) - Pi) = 1.4201994774470...

Mathematica

CoefficientList[Series[1/(Cos[x]-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)

Prog

(PARI) {a(n)=n!*polcoeff(1/(cos(x+x*O(x^n))-x), n)}
(PARI) x='x+O('x^30); Vec(serlaplace(1/(cos(x) - x))) \\ G. C. Greubel, Jul 10 2018
(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

Crossrefs

Cf. A006154, A003957.

Sequence in context: A111262 A139134 A216373 * A256124 A109577 A368265

Adjacent sequences: A200306 A200307 A200308 * A200310 A200311 A200312

Keyword

nonn

Author

Paul D. Hanna, Nov 15 2011

Status

approved