%I #8 Jan 26 2014 12:13:35
%S 1,2,11,100,1269,20680,411655,9681040,262644825,8074470560,
%T 277410402675,10533370203200,438024379604525,19798139730512000,
%U 966408931916064975,50666524133429152000,2839464166814487200625,169393547843598544960000,10717798206604377757886875
%N E.g.f. A(x) satisfies: A( cos(x) - exp(-x) ) = x.
%F a(n) ~ n^(n-1) / (sqrt(cos(s)+sin(s)) * exp(n) * (cos(s)-sin(s))^(n-1/2)), where s = 0.5885327439818610774... is the root of the equation sin(s) = exp(-s). - _Vaclav Kotesovec_, Jan 26 2014
%e E.g.f.: A(x) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1269*x^5/5! +...
%e The series reversion of the e.g.f. begins:
%e cos(x) - exp(-x) = x - 2*x^2/2! + x^3/3! + x^5/5! - 2*x^6/6! + x^7/7! + x^9/9! - 2*x^10/10! + x^11/11! + x^13/13! - 2*x^14/14! +...
%t Rest[CoefficientList[InverseSeries[Series[Cos[x] - E^(-x),{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 26 2014 *)
%o (PARI) {a(n)=n!*polcoeff(serreverse(cos(x+x^2*O(x^n))-exp(-x+x^2*O(x^n))),n)}
%K nonn
%O 1,2
%A _Paul D. Hanna_, Feb 02 2012
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