OFFSET
0,2
COMMENTS
Limit (a(n)/n!)^(-1/n) = r = 0.516421075467358505089... where exp(r) = tan(2*r).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
E.g.f. satisfies: exp(x) = (cos(2*x) + sin(2*x)*A(x)) / (cos(2*x)*A(x) - sin(2*x)).
a(n) ~ 2*n! / ((4-sin(4*r)) * r^(n+1)), where r is described above. - Vaclav Kotesovec, Jul 29 2014
EXAMPLE
E.g.f.: A(x) = 1 + 3*x + 9*x^2/2! + 55*x^3/3! + 417*x^4/4! + 4063*x^5/5! +...
Note that the logarithm of the e.g.f. is an odd function:
log(A(x)) = 3*x + 28*x^3/3! + 1300*x^5/5! + 146908*x^7/7! + 30865700*x^9/9! +...
thus A(x)*A(-x) = 1.
MATHEMATICA
CoefficientList[Series[(Cos[2*x] + Sin[2*x]*E^x) / (Cos[2*x]*E^x - Sin[2*x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jul 29 2014 *)
PROG
(PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); A=(cos(2*X)+sin(2*X)*exp(X))/(cos(2*X)*exp(X)-sin(2*X)); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 28 2014
STATUS
approved