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A194957 E.g.f. satisfies: A(x) = cos(x) + x*A(x)^2.

%I A194957 #16 Mar 29 2017 08:54:51
%S A194957 1,1,3,24,265,3880,71099,1568896,40538065,1201214592,40167734579,
%T A194957 1496537977088,61482321148249,2761475028475904,134626530031186283,
%U A194957 7080317004476928000,399585298475488156321,24087998749960900673536,1544790381323645227703651
%N A194957 E.g.f. satisfies: A(x) = cos(x) + x*A(x)^2.
%C A194957 The radius of convergence r of the e.g.f. A(x) satisfies: r = 1/(4*cos(r)) = limit (n+1)*a(n)/a(n+1) = 0.2585985822541... with A(r) = 1/(2*r) = 1.933498612565961...
%H A194957 G. C. Greubel, <a href="/A194957/b194957.txt">Table of n, a(n) for n = 0..365</a>
%F A194957 E.g.f.: A(x) = (1 - sqrt(1 - 4*x*cos(x))) / (2*x).
%F A194957 a(n) = (-1)^floor(n/2)*(1 + (-1)^n)/2 + n*Sum_{k=0..n-1} binomial(n-1,k)*a(k)*a(n-1-k) for n >= 0.
%F A194957 a(n) ~ n! * sqrt(cos(r)-sin(r)*r)/(2*sqrt(Pi)*r^(n+1/2)*n^(3/2)), where r = 0.258598582254189... is defined in the comment. - _Vaclav Kotesovec_, Sep 21 2013
%e A194957 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 24*x^3/3! + 265*x^4/4! + 3880*x^5/5! + ...
%e A194957 Related expansion:
%e A194957 A(x)^2 = 1 + 2*x + 8*x^2/2! + 66*x^3/3! + 776*x^4/4! + 11850*x^5/5! + ...
%t A194957 CoefficientList[Series[(1-Sqrt[1-4*x*Cos[x]])/(2*x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 21 2013 *)
%o A194957 (PARI) {a(n)=n!*polcoeff((1 - sqrt(1 - 4*x*cos(x +O(x^(n+2))))) / (2*x),n)}
%o A194957 (PARI) {a(n)=(-1)^(n\2)*(1 + (-1)^n)/2 +n*sum(k=0, n-1, binomial(n-1, k)*a(k)*a(n-1-k))}
%Y A194957 Cf. A194958, A194471.
%K A194957 nonn
%O A194957 0,3
%A A194957 _Paul D. Hanna_, Sep 06 2011