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A143600 E.g.f. satisfies: A(x) = exp(x*A(x)/A(-x)).

%I A143600 #23 Jul 19 2016 11:21:15
%S A143600 1,1,5,25,249,2561,40573,641817,14110001,302279617,8530496181,
%T A143600 230851019609,7964867290537,260618470319169,10635790073585069,
%U A143600 408342804482252761,19246730825243728737,848289638051491455617,45356940470607637151845,2257054105205570995111833
%N A143600 E.g.f. satisfies: A(x) = exp(x*A(x)/A(-x)).
%H A143600 Vaclav Kotesovec, <a href="/A143600/b143600.txt">Table of n, a(n) for n = 0..175</a>
%F A143600 E.g.f. A(x) satisfies:
%F A143600 (1) A(x) = exp(x*exp(2x*G(2x))) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
%F A143600 (2) [A(x)/A(-x) + A(-x)/A(x)]/2 = G(2x) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
%F A143600 (3) A(x)/A(-x) = exp(x*[A(x)/A(-x) + A(-x)/A(x)]) = F(2x) where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
%F A143600 (4) A(x) = Sum_{n>=0} (n+1)^(n-1) * x^n/n! / A(-x)^n.
%F A143600 (5) A(x)^m = Sum_{n>=0} m*(n+m)^(n-1) * x^n/n! / A(-x)^n.
%F A143600 (6) log(A(x)) = Sum_{n>=1} n^(n-1) * x^n/n! / A(-x)^n = x*A(x)/A(-x).
%F A143600 Formulas (4), (5), and (6) are due to LambertW identities. - _Paul D. Hanna_, Nov 05 2012
%F A143600 a(n) ~ c * n! / (n^(3/2) * r^n), where r = 0.33137170967459079... is the root of the equation sqrt(1+4*r^2) = log((1+sqrt(1+4*r^2))/(2*r)), and c = 1.35397895306096963692514418... if n is even, and c = 1.281887793570420328585518150... if n is odd. - _Vaclav Kotesovec_, Feb 25 2014
%e A143600 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! +...
%e A143600 A LambertW identity yields the series:
%e A143600 A(x) = 1 + x/A(-x) + 3^1*x^2/2!/A(-x)^2 + 4^2*x^3/3!/A(-x)^3 + 5^3*x^4/4!/A(-x)^4 + 6^4*x^5/5!/A(-x)^5 +...+ (n+1)^(n-1)*x^n/n!/A(-x)^n +...
%e A143600 RELATED EXPANSIONS.
%e A143600 A(x)/A(-x) = F(2x) where F(x) is the e.g.f. of A058014:
%e A143600 A(x)/A(-x) = 1 + 2*x + 4*x^2/2! + 32*x^3/3! + 208*x^4/4! + 3072*x^5/5! +...
%e A143600 F(x) = 1 + x + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +...
%e A143600 which satisfies: F(x) = exp(x*(F(x) + 1/F(x))/2).
%e A143600 (A(x)/A(-x) + A(-x)/A(x))/2 = G(2x) where G(x) is the e.g.f. of A143601:
%e A143600 (A(x)/A(-x) + A(-x)/A(x))/2 = 1 + 4*x^2/2! + 208*x^4/4! + 34624*x^6/6! +...
%e A143600 G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
%e A143600 which satisfies G(x) = cosh(x*G(x)).
%o A143600 (PARI) a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x*A/subst(A,x,-x)));n!*polcoeff(A,n)
%o A143600 (PARI) /* Formula Using a LambertW Identity: */
%o A143600 {a(n)=local(A=1);for(i=1,n,A=sum(k=0,n,(k+1)^(k-1)*x^k/k!/subst(A,x,-x)^k+x*O(x^n)));n!*polcoeff(A,n)}
%o A143600 for(n=0,25,print1(a(n),", ")) \\ _Paul D. Hanna_, Nov 05 2012
%Y A143600 Cf. A058014, A143601, A007106.
%K A143600 nonn
%O A143600 0,3
%A A143600 _Paul D. Hanna_, Aug 26 2008