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

0,2

Comments

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

Links

Table of n, a(n) for n=0..24.

Example

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.

Prog

(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)}

Crossrefs

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

Sequence in context: A006191 A149959 A161941 * A149960 A149961 A141754

Adjacent sequences: A120896 A120897 A120898 * A120900 A120901 A120902

Keyword

nonn

Author

Paul D. Hanna, Jul 14 2006

Status

approved