|
%I #12 Aug 01 2014 06:36:01
%S 1,3,9,55,417,4063,47089,638815,9893057,172430463,3338825169,
%T 71119417375,1652583598497,41600978876863,1127787743366449,
%U 32757802728304735,1014917534672868737,33409942701676459263,1164512820203174614929,42844385538424750452895,1659281060494160725575777
%N E.g.f.: (cos(2*x) + sin(2*x)*exp(x)) / (cos(2*x)*exp(x) - sin(2*x)).
%C Limit (a(n)/n!)^(-1/n) = r = 0.516421075467358505089... where exp(r) = tan(2*r).
%H Vincenzo Librandi, <a href="/A245117/b245117.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f. satisfies: exp(x) = (cos(2*x) + sin(2*x)*A(x)) / (cos(2*x)*A(x) - sin(2*x)).
%F a(n) ~ 2*n! / ((4-sin(4*r)) * r^(n+1)), where r is described above. - _Vaclav Kotesovec_, Jul 29 2014
%e E.g.f.: A(x) = 1 + 3*x + 9*x^2/2! + 55*x^3/3! + 417*x^4/4! + 4063*x^5/5! +...
%e Note that the logarithm of the e.g.f. is an odd function:
%e log(A(x)) = 3*x + 28*x^3/3! + 1300*x^5/5! + 146908*x^7/7! + 30865700*x^9/9! +...
%e thus A(x)*A(-x) = 1.
%t 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 *)
%o (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)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A245116.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 28 2014
| 