%I #8 Feb 26 2014 09:01:45
%S 1,1,1,7,49,431,5281,69735,1163681,21106783,443495041,10238783303,
%T 261253326577,7289538431631,220029116061665,7180115829246391,
%U 250782957484550977,9365723147426913215,371694221511705312769,15638233703457023744679
%N E.g.f. satisfies: A(x) = exp(x/A(x)) - exp(x) + exp(x*A(x)).
%H Vaclav Kotesovec, <a href="/A195514/b195514.txt">Table of n, a(n) for n = 0..280</a>
%F E.g.f. satisfies:
%F _ A(x) = Sum_{n>=0} x^n*(1/A(x)^n - 1 + A(x)^n) / n!.
%F _ A(x) = B(x/A(x)) where B(x) = A(x*B(x)) equals the e.g.f. of A195512 and satisfies: B(x) = exp(x) - exp(x*B(x)) + exp(x*B(x)^2).
%F _ A(x) = C(x/A(x)^2) where C(x) = A(x*C(x)^2) equals the e.g.f. of A195513 and satisfies: C(x) = exp(x*C(x)) - exp(x*C(x)^2) + exp(x*C(x)^3).
%F a(n) ~ sqrt((s^3*(exp(r/s) - exp(r)*s + exp(r*s)*s^2)) / (exp(r*s)*r*s^4 + exp(r/s)*(r + 2*s))) * n^(n-1) / (exp(n) * r^n), where r = 0.4188500786493484757126061423... and s = 2.29528345389537775167313... are roots of system of the equations exp(r) + s = exp(r/s) + exp(r*s), exp(r*s)*r = 1 + (exp(r/s)*r)/s^2. - _Vaclav Kotesovec_, Feb 26 2014
%e E.g.f.: A(x) = 1 + x + x^2/2! + 7*x^3/3! + 49*x^4/4! + 431*x^5/5! +...
%e Related series begin:
%e exp(x/A(x)) = 1 + x - x^2/2! - 2*x^3/3! - 15*x^4/4! - 124*x^5/5! -...
%e exp(x*A(x)) = 1 + x + 3*x^2/2! + 10*x^3/3! + 65*x^4/4! + 556*x^5/5! +...
%e The e.g.f. of A195512 begins:
%e B(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4426*x^5/5! +...
%e where A(x*B(x)) = B(x) = exp(x) - exp(x*B(x)) + exp(x*B(x)^2).
%e The e.g.f. of A195513 begins:
%e C(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 969*x^4/4! + 23471*x^5/5! +...
%e where A(x*C(x)^2) = C(x) = exp(x*C(x)) - exp(x*C(x)^2) + exp(x*C(x)^3).
%o (PARI) {a(n)=local(X=x+x*O(x^n),A=1+X);for(i=1,n,A=exp(X/A)-exp(X)+exp(X*A));n!*polcoeff(A,n)}
%o (PARI) {a(n)=local(X=x+x*O(x^n),A=1+X);for(i=1,n,A=sum(m=0,n,x^m*(1/A^m-1+A^m)/m!));n!*polcoeff(A,n)}
%Y Cf. A195512, A195513.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Sep 20 2011