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A194453 E.g.f. satisfies: A(x) = exp(x) - sqrt(1 - A(x)^2).

%I A194453 #23 Jun 27 2022 01:17:52
%S A194453 1,2,7,44,421,5342,83707,1556984,33495721,817880282,22341817807,
%T A194453 675009140324,22347321835021,804481291160822,31286388389010307,
%U A194453 1307157133950142064,58390601701376026321,2776992745284738150962,140092142842449580093207
%N A194453 E.g.f. satisfies: A(x) = exp(x) - sqrt(1 - A(x)^2).
%C A194453 Compare e.g.f. to the identity: cosh(x) = exp(x) - sqrt(cosh(x)^2 - 1).
%H A194453 Vincenzo Librandi, <a href="/A194453/b194453.txt">Table of n, a(n) for n = 1..200</a>
%F A194453 E.g.f.: A(x) = (exp(x) - sqrt(2 - exp(2*x))) / 2.
%F A194453 E.g.f. A(x) satisfies: A( log(sqrt(1-x^2) + x) ) = x; thus, e.g.f. A(x) is a signed series reversion of the e.g.f. of A194349.
%F A194453 E.g.f. A(x) satisfies: A(x) = sinh(x) + exp(-x)*A(x)^2. - _Paul D. Hanna_, Aug 29 2018
%F A194453 a(n) ~ (2*n)^(n-1) / (exp(n) * log(2)^(n-1/2)). - _Vaclav Kotesovec_, Sep 22 2013
%e A194453 E.g.f.: A(x) = x + 2*x^2/2! + 7*x^3/3! + 44*x^4/4! + 421*x^5/5! +...
%e A194453 where A( log(sqrt(1-x^2) + x) ) = x and
%e A194453 log(sqrt(1-x^2) + x) = x - 2*x^2/2! + 5*x^3/3! - 24*x^4/4! + 129*x^5/5! - 960*x^6/6! +...+ -(-1)^n*A194349(n)*x^n/n! +...
%p A194453 a:= n-> n!*coeff(series(RootOf(A=exp(x)-sqrt(1-A^2), A), x, n+1), x, n):
%p A194453 seq(a(n), n=1..20);  # _Alois P. Heinz_, Sep 21 2013
%t A194453 Rest[CoefficientList[Series[(E^x-Sqrt[2-E^(2*x)])/2, {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Sep 22 2013 *)
%o A194453 (PARI) {a(n)=n!*polcoeff(serreverse( log(sqrt(1-x^2 +O(x^(n+2)))+x)),n)}
%o A194453 (PARI) {a(n)=n!*polcoeff((exp(x+x*O(x^n))-sqrt(2-exp(2*x+x*O(x^n))))/2,n)}
%Y A194453 Cf. A194349.
%Y A194453 Partial sums of A352117.
%K A194453 nonn
%O A194453 1,2
%A A194453 _Paul D. Hanna_, Aug 24 2011