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%I #14 Aug 01 2014 06:36:28
%S 1,1,1,3,9,31,141,823,5009,32671,252181,2203143,20062809,194886511,
%T 2105627421,24559293463,298779205409,3849283295551,53331796237861,
%U 780608097567783,11900618940664809,190559301996683791,3219596479104915501,56821283929863042103,1041441834338061113009
%N E.g.f.: (cos(x) + sin(x)*exp(x)) / (cos(x)*exp(x) - sin(x)).
%C Limit (a(n)/n!)^(-1/n) = r = 1.306326940423079236174... where exp(r) = tan(r).
%H Vincenzo Librandi, <a href="/A245116/b245116.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f. satisfies: exp(x) = (cos(x) + sin(x)*A(x)) / (cos(x)*A(x) - sin(x)).
%F a(n) ~ 2*n! / ((2-sin(2*r)) * r^(n+1)), where r is described above. - _Vaclav Kotesovec_, Jul 29 2014
%e 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! +...
%e Note that the logarithm of the e.g.f. is an odd function:
%e 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! +...
%e thus A(x)*A(-x) = 1.
%t 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 *)
%o (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)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A245117.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jul 28 2014
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