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

1, 0, 1, 2, 6, 22, 100, 554, 3654, 28014, 244572, 2392042, 25877610, 306553246, 3944541224, 54764396346, 815786104186, 12976263731454, 219490418886728, 3933636232278866, 74453982353188846, 1484056255756797222, 31071499784792496588, 681729867750992165514, 15641641334118250802462

Offset: 0

Author

Paul D. Hanna, Jul 24 2014

Keywords

• nonn

Examples

			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 +...
where the logarithmic derivative equals (A(x) - 1 - x^2)/x^2:
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 +...
thus the logarithm begins:
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 +...
		

References

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

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A245308.

Programs

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)}
for(n=0,30,print1(a(n),", "))

PARI

/* From A(x) = exp(-x)*G(x), where G(x) = e.g.f. of A245308: */
{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))));
polcoeff(exp(-x +x*O(x^n))*G,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(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.

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

a(n) ~ BesselJ(1,2) * (n-1)!. - Vaclav Kotesovec, Jul 25 2014