A154677 G.f. satisfies: A(x/A(x)) = G(x) where G(x) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan numbers).

1, 1, 3, 13, 70, 440, 3116, 24274, 204407, 1836339, 17425275, 173329307, 1796783304, 19323703019, 214843877103, 2462522274426, 29032815570544, 351447240945518, 4361579736404011, 55424256247911490, 720399315622779670, 9569215299494074698, 129799982362958621827

Offset: 0

Author

Paul D. Hanna, Jan 14 2009

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 70*x^4 + 440*x^5 + ... where
A(x/A(x)) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + ...
x/A(x) = x - x^2 - 2*x^3 - 8*x^4 - 43*x^5 - 277*x^6 - 2026*x^7 - ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000108.

Cf. variants: A168448, A168478. - Paul D. Hanna, Dec 06 2009

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)))));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))));polcoeff(A,n)} \\ Paul D. Hanna, Dec 06 2009

Formula

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

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

G.f. satisfies: A(x) = 1 + A(x)^2*Series_Reversion(x/A(x)). - Paul D. Hanna, Dec 06 2009