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A245119 G.f. satisfies: A(x) = 1 + x^2 + x^2*A'(x)/A(x).

%I A245119 #12 Jul 25 2014 03:07:49
%S A245119 1,0,1,2,6,22,100,554,3654,28014,244572,2392042,25877610,306553246,
%T A245119 3944541224,54764396346,815786104186,12976263731454,219490418886728,
%U A245119 3933636232278866,74453982353188846,1484056255756797222,31071499784792496588,681729867750992165514,15641641334118250802462
%N A245119 G.f. satisfies: A(x) = 1 + x^2 + x^2*A'(x)/A(x).
%D A245119 Compare g.f. to: G(x) = 1 + x + x^2*G'(x)/G(x) when G(x) = 1/(1-x).
%H A245119 Paul D. Hanna, <a href="/A245119/b245119.txt">Table of n, a(n) for n = 0..300</a>
%F A245119 G.f. A(x) satisfies:
%F A245119 (1) A(x) = exp(-x)*G(x) where G(x) = exp(x)*(1 + x^2*G'(x)/G(x)) is the e.g.f. of A245308.
%F A245119 (2) A(x) = exp( Integral (A(x) - 1 - x^2)/x^2 dx ).
%F A245119 a(n) ~ BesselJ(1,2) * (n-1)!. - _Vaclav Kotesovec_, Jul 25 2014
%e A245119 G.f.: A(x) = 1 + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 100*x^6 + 554*x^7 + 3654*x^8 +...
%e A245119 where the logarithmic derivative equals (A(x) - 1 - x^2)/x^2:
%e A245119 A'(x)/A(x) = 2*x + 6*x^2 + 22*x^3 + 100*x^4 + 554*x^5 + 3654*x^6 +...+ a(n+2)*x^n +...
%e A245119 thus the logarithm begins:
%e A245119 log(A(x)) = 2*x^2/2 + 6*x^3/3 + 22*x^4/4 + 100*x^5/5 + 554*x^6/6 + 3654*x^7/7 +...+ a(n+1)*x^n/n +...
%o A245119 (PARI) {a(n)=local(A=1+x^2);for(i=1,n,A = 1 + x^2 + x^2*A'/(A +x*O(x^n)));polcoeff(A,n)}
%o A245119 for(n=0,30,print1(a(n),", "))
%o A245119 (PARI) /* From A(x) = exp(-x)*G(x), where G(x) = e.g.f. of A245308: */
%o A245119 {a(n)=local(G=1+x);for(i=1,n,G = exp(x +x*O(x^n))*(1 + x^2*G'/(G +x*O(x^n))));
%o A245119 polcoeff(exp(-x +x*O(x^n))*G,n)}
%o A245119 for(n=0,30,print1(a(n),", "))
%Y A245119 Cf. A245308.
%K A245119 nonn
%O A245119 0,4
%A A245119 _Paul D. Hanna_, Jul 24 2014