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%I #11 Feb 13 2017 03:35:45
%S 1,1,5,36,409,6280,121501,2839424,77834737,2449360512,87040440181,
%T 3447798906112,150645874207753,7197909122453504,373365727806824845,
%U 20895734364795187200,1255062315134651501281,80528111291313595580416,5497183726333878664852453
%N E.g.f. satisfies: A(x) = cosh(x) + x*A(x)^2.
%C The radius of convergence r of the e.g.f. A(x) satisfies: r = 1/(4*cosh(r)) = limit (n+1)*a(n)/a(n+1) = 0.24280736240... with A(r) = 1/(2*r) = 2.059245630...
%H G. C. Greubel, <a href="/A194958/b194958.txt">Table of n, a(n) for n = 0..360</a>
%F E.g.f.: A(x) = (1 - sqrt(1 - 4*x*cosh(x))) / (2*x).
%F a(n) = (1 + (-1)^n)/2 + n*Sum_{k=0..n-1} C(n-1,k)*a(k)*a(n-1-k) for n>=0.
%F a(n) ~ n! * sqrt(cosh(r)+r*sinh(r))/(2*r^(n+1/2)*sqrt(Pi)*n^(3/2)), where r = 0.2428073624... is defined in the comment. - _Vaclav Kotesovec_, Sep 21 2013
%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 36*x^3/3! + 409*x^4/4! + 6280*x^5/5! +...
%e Related expansion:
%e A(x)^2 = 1 + 2*x + 12*x^2/2! + 102*x^3/3! + 1256*x^4/4! + 20250*x^5/5! +...
%t CoefficientList[Series[(1-Sqrt[1-4*x*Cosh[x]])/(2*x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 21 2013 *)
%o (PARI) {a(n)=n!*polcoeff((1 - sqrt(1 - 4*x*cosh(x +O(x^(n+2))))) / (2*x),n)}
%o (PARI) {a(n)=(1 + (-1)^n)/2 +n*sum(k=0, n-1, binomial(n-1, k)*a(k)*a(n-1-k))}
%Y Cf. A194957, A194471.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 06 2011
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