A245117 E.g.f.: (cos(2*x) + sin(2*x)*exp(x)) / (cos(2*x)*exp(x) - sin(2*x)).

1, 3, 9, 55, 417, 4063, 47089, 638815, 9893057, 172430463, 3338825169, 71119417375, 1652583598497, 41600978876863, 1127787743366449, 32757802728304735, 1014917534672868737, 33409942701676459263, 1164512820203174614929, 42844385538424750452895, 1659281060494160725575777

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

Cf. A245116.

Sequence in context: A032179 A233189 A175117 * A183303 A075979 A128681

Adjacent sequences: A245114 A245115 A245116 * A245118 A245119 A245120

Keyword

nonn

Author

Paul D. Hanna, Jul 28 2014

Status

approved