OFFSET
0,2
COMMENTS
Sum_{n>=0} a(n)x^n/n! = exp(x)/(cos(x) - sin(x)).
FORMULA
Sum_{n>=0} a(n)x^n/n! = exp(x)/(cos(x) - sin(x)).
exp(x)/(cos(x) - sin(x)) = 1/G(0); G(k) = 1-2*x/(4*k+1+x*(4*k+1)/(2*(2*k+1)-x-2*(x^2)*(2*k+1)/((x^2)-(2*k+2)*(4*k+3)/G(k+1)))); (continued fraction).
G.f.: 1/G(0) where G(k) = 1 - 2*x*(k+1) - 2*x^2*(k+1)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 12 2013
a(n) ~ n! * exp(Pi/4)*2^(2*n+3/2)/Pi^(n+1). - Vaclav Kotesovec, Jun 27 2013
G.f.: T(0)/(1-2*x), where T(k) = 1 - 2*x^2*(k+1)^2/(2*x^2*(k+1)^2 - (1 - 2*x - 2*x*k)*(1 - 4*x - 2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 29 2013
MAPLE
A:=series(exp(x)/(cos(x)-sin(x)), x, 40);
G(x):=A : f[0]:=G(x): for n from 1 to 41 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..40);
MATHEMATICA
nn = 30; Range[0, nn]! CoefficientList[Series[Exp[x]/(Cos[x] - Sin[x]), {x, 0, nn}], x] (* T. D. Noe, Dec 05 2011 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Sergei N. Gladkovskii, Nov 27 2011
STATUS
approved