A245119 - OEIS

%I #12 Jul 25 2014 03:07:49

%S 1,0,1,2,6,22,100,554,3654,28014,244572,2392042,25877610,306553246,

%T 3944541224,54764396346,815786104186,12976263731454,219490418886728,

%U 3933636232278866,74453982353188846,1484056255756797222,31071499784792496588,681729867750992165514,15641641334118250802462

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

%D Compare g.f. to: G(x) = 1 + x + x^2*G'(x)/G(x) when G(x) = 1/(1-x).

%H Paul D. Hanna, <a href="/A245119/b245119.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) satisfies:

%F (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 (2) A(x) = exp( Integral (A(x) - 1 - x^2)/x^2 dx ).

%F a(n) ~ BesselJ(1,2) * (n-1)!. - _Vaclav Kotesovec_, Jul 25 2014

%e 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 where the logarithmic derivative equals (A(x) - 1 - x^2)/x^2:

%e 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 thus the logarithm begins:

%e 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 (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 for(n=0,30,print1(a(n),", "))

%o (PARI) /* From A(x) = exp(-x)*G(x), where G(x) = e.g.f. of A245308: */

%o {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 polcoeff(exp(-x +x*O(x^n))*G,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A245308.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jul 24 2014