Reference request: Catalan number of type B

Question

Are there some generalized Catalan number of type $B$ such that the sequence of the numbers is $3,9,29,97,333$ for $n=2,3,4,5,6$?

As discussed in this previous question, there are at least two types of generalizations of Catalan numbers for Coxeter groups:

(1) "A uniform bijection between nonnesting and noncrossing partitions" by Armstrong, Stump, and Thomas.

(2) "The enumeration of fully commutative elements of coxeter Groups" by Stembridge.

But they do not give this sequence. (2) gives $7,24,83,293,1055$ for $n=2,3,4,5,6$. Thank you very much.

Answer 1

This series $a_n=1, 3, 9, 29, 97, 333, 1165, 4135, 14845, 53791, 196417$ is recorded as the p-INVERT of the series of Catalan numbers $c_n$, for the polynomial $p(S)=1-S-S^2$.

$$\sum_{n=0}^\infty a_n x^n=\frac{1}{x}\left(-1+\frac{1}{p\left(\sum_{n=1}^\infty c_{n-1} x^n\right)}\right)$$

The corresponding relation to the Catalan numbers $b_n$ of type B is

$$\sum_{n=0}^\infty a_n x^n=\frac{1}{x}\left(-1+\frac{1}{p\left(\sum_{n=1}^\infty b_{n-1} x^n/n\right)}\right)$$