A120899 G.f. satisfies: A(x) = C(x)^2 * A(x^3*C(x)^4), where C(x) is the g.f. of the Catalan numbers (A000108).

1, 2, 5, 16, 54, 186, 654, 2338, 8463, 30938, 114022, 423096, 1579049, 5922512, 22309350, 84354388, 320020227, 1217689680, 4645693038, 17766596202, 68092473570, 261486788434, 1005962436536, 3876412305114, 14960183283203

Offset: 0

Author

Paul D. Hanna, Jul 14 2006

Keywords

• nonn

Comments

Column 0 of triangle A120898 (cascadence of 1+2x+x^2). Self-convolution of A120900.

Examples

			A(x) = 1 + 2*x + 5*x^2 + 16*x^3 + 54*x^4 + 186*x^5 + 654*x^6 +...
= C(x)^2 * A(x^3*C(x)^4) where
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
is the g.f. of the Catalan numbers (A000108): C(x) = 1 + x*C(x)^2.
		

Crossrefs

Cf. A120898, A120900 (square-root), A120901, A120902; A000108; variants: A092684, A092687, A120895.

Programs

PARI

{a(n)=local(A=1+x,C=(1/x*serreverse(x/(1+2*x+x^2+x*O(x^n))))^(1/2)); for(i=0,n,A=C^2*subst(A,x,x^3*C^4 +x*O(x^n)));polcoeff(A,n,x)}