A168448 G.f. satisfies: A(x/A(x)^2) = C(x) where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

1, 1, 4, 26, 226, 2395, 29278, 398499, 5899534, 93507783, 1569405110, 27672405800, 509622262860, 9759305238932, 193673399146066, 3972141366536794, 84010899306559470, 1829057795368804875, 40931310532585505770, 940322157062673670051, 22152626055397162566438

Offset: 0

Author

Paul D. Hanna, Dec 06 2009

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 226*x^4 + 2395*x^5 +...
A(x/A(x)^2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
		

Crossrefs

Cf. A154677, A168449.

Programs

PARI

{a(n)=local(A=1+x, F=sum(k=0, n, binomial(2*k+1, k)/(2*k+1)*x^k)+x*O(x^n)); for(i=0, n, A=subst(F, x, serreverse(x/(A+x*O(x^n))^2))); polcoeff(A, n)}

PARI

{a(n)=local(A=1+x);for(i=1,n,A=1+A^2*serreverse(x/(A+x*O(x^n))^2)); polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = 1 + A(x)^2*Series_Reversion[x/A(x)^2].

G.f. satisfies: A( (x-x^2)/A(x-x^2)^2 ) = 1/(1-x).

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