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A195510 E.g.f. satisfies: A(x) = (exp(x*A(x)) + exp(x/A(x)))/2.

%I A195510 #12 Jan 11 2014 10:37:16
%S A195510 1,1,1,4,25,156,1561,15968,205521,2996416,48245521,891169632,
%T A195510 17701061929,390008578688,9273243108585,237666643184896,
%U A195510 6557636429111329,192526245072832512,6029011794295629217,199832740470058237952,7000664928137597341881
%N A195510 E.g.f. satisfies: A(x) = (exp(x*A(x)) + exp(x/A(x)))/2.
%H A195510 Vaclav Kotesovec, <a href="/A195510/b195510.txt">Table of n, a(n) for n = 0..100</a>
%F A195510 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * (A(x)^n + 1/A(x)^n)/2.
%F A195510 E.g.f. satisfies: A(x) = B(x/A(x)) where B(x) = A(x*B(x)) is the g.f. of A195511 and satisfies: B(x) = (exp(x) + exp(x*B(x)^2))/2.
%F A195510 a(n) ~ sqrt(s^3*(-1+2*r*s+s^2)/(-2*s + r^2*s*(1+s^2) + r*(-1+2*s^2+s^4))) * n^(n-1) / (exp(n) * r^n), where r = 0.53064794557035571... and s = 2.58970226260372420623... are the roots of the equations exp(r/s) + exp(r*s) = 2*s, exp(r*s)*r*s = exp(r/s)*r/s + 2*s. - _Vaclav Kotesovec_, Jan 11 2014
%e A195510 E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 156*x^5/5! +...
%e A195510 Related series begin:
%e A195510 exp(x*A(x)) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 376*x^5/5! + 3247*x^6/6! +...
%e A195510 exp(x/A(x)) = 1 + x - x^2/2! - 2*x^3/3! - 3*x^4/4! - 64*x^5/5! - 125*x^6/6! -...
%e A195510 The g.f. B(x) of A195511 begins:
%e A195510 B(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 197*x^4/4! + 2801*x^5/5! + 50407*x^6/6! +...
%e A195510 where A(x*B(x)) = B(x) = (exp(x) + exp(x*B(x)^2))/2.
%o A195510 (PARI) {a(n)=local(X=x+x*O(x^n),A=1+X);for(i=1,n,A=(exp(X*A)+exp(X/A))/2);n!*polcoeff(A,n)}
%Y A195510 Cf. A195511.
%K A195510 nonn
%O A195510 0,4
%A A195510 _Paul D. Hanna_, Sep 19 2011