The ubiquitous Catalan numbers \$C_n\$ count the number of Dyck paths, sequences of up-steps and down-steps of length \$2n\$ that start and end on a horizontal line and never go below said line. Many other interesting sequences can be defined as the number of Dyck paths satisfying given conditions, of which the Fine sequence \$F_n\$ (not the Fibonacci numbers and not related to any common definition of "fine") is one.
Let a hill be a sequence of an up-step followed by a down-step that starts – and therefore also ends – on the horizontal line. \$F_n\$ is then the number of Dyck paths of length \$2n\$ with no hills. The picture below illustrates this: there are \$C_5=42\$ Dyck paths of length \$10\$, of which \$F_5=18\$ (marked in black) have no hills.
This sequence is OEIS A000957 and begins $$\begin{array}{c|ccccccccccc} n&0&1&2&3&4&5&6&7&8&9&10\\ \hline F_n&1&0&1&2&6&18&57&186&622&2120&7338 \end{array}$$ $$\begin{array}{c|ccccccccccc} n&11&12&13&14&15\\ \hline F_n&25724&91144&325878&1174281&4260282 \end{array}$$
Other things counted by the Fine numbers include
- the number of Dyck paths of length \$2n\$ beginning with an even number of up-steps
- the number of ordered trees with \$n+1\$ vertices where the root has no leaf children
- the number of ordered trees with \$n+1\$ vertices where the root has an even number of children
- and so on. For more interpretations see Deutsch and Shapiro's "A survey of the Fine numbers".
Formulas
You may use any correct formula to generate the sequence. Here are some:
- The generating function is $$\sum_{n=0}^\infty F_nz^n=\frac1z\cdot\frac{1-\sqrt{1-4z}}{3-\sqrt{1-4z}}$$
- For \$n\ge1\$, \$C_n=2F_n+F_{n-1}\$.
- An explicit formula: $$F_n=\frac1{n+1}\sum_{k=0}^n(-1)^k(k+1)\binom{2n-k}{n-k}$$
Task
Standard sequence rules apply to this challenge, where permissible behaviours are
- outputting the \$n\$th term in 0- or 1-based indexing given \$n\$
- outputting the first \$n\$ terms given \$n\$
- outputting the infinite sequence with no input, either by printing or returning a lazy list/generator
This is code-golf; fewest bytes wins.