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A200309
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Expansion of e.g.f.: 1/(cos(x) - x).
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4
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1, 1, 3, 12, 65, 440, 3571, 33824, 366113, 4458240, 60321091, 897774592, 14576528801, 256391130112, 4856647308787, 98567413125120, 2133825372539585, 49080991762153472, 1195339768057071619, 30729146849826701312, 831545527540481198465, 23627123985544955559936
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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...
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MATHEMATICA
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CoefficientList[Series[1/(Cos[x]-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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