A245116 E.g.f.: (cos(x) + sin(x)*exp(x)) / (cos(x)*exp(x) - sin(x)).

1, 1, 1, 3, 9, 31, 141, 823, 5009, 32671, 252181, 2203143, 20062809, 194886511, 2105627421, 24559293463, 298779205409, 3849283295551, 53331796237861, 780608097567783, 11900618940664809, 190559301996683791, 3219596479104915501, 56821283929863042103, 1041441834338061113009

Offset

0,4

Comments

Limit (a(n)/n!)^(-1/n) = r = 1.306326940423079236174... where exp(r) = tan(r).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

E.g.f. satisfies: exp(x) = (cos(x) + sin(x)*A(x)) / (cos(x)*A(x) - sin(x)).

a(n) ~ 2*n! / ((2-sin(2*r)) * r^(n+1)), where r is described above. - Vaclav Kotesovec, Jul 29 2014

Example

E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 31*x^5/5! + 141*x^6/6! +...
Note that the logarithm of the e.g.f. is an odd function:
log(A(x)) = x + 2*x^3/3! + 10*x^5/5! + 262*x^7/7! + 6130*x^9/9! + 433022*x^11/11! + 26718250*x^13/13! + 3408852982*x^15/15! +...
thus A(x)*A(-x) = 1.

Mathematica

CoefficientList[Series[(Cos[x] + Sin[x]*E^x) / (Cos[x]*E^x - Sin[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(X)+sin(X)*exp(X))/(cos(X)*exp(X)-sin(X)); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A245117.

Sequence in context: A090595 A027040 A111063 * A255382 A089475 A351185

Adjacent sequences: A245113 A245114 A245115 * A245117 A245118 A245119

Keyword

nonn

Author

Paul D. Hanna, Jul 28 2014

Status

approved