Jonah Formula

A formula for the generalized Catalan Number ${}_pd_{qi}$. The general formula is

$${n-q\choose k-1}=\sum_{i=1}^k {}_pd_{qi}{n-pi\choose k-i},$$

where ${n\choose k}$ is a Binomial Coefficient, although Jonah's original formula corresponded to $p=2$, $q=0$ (Hilton and Pederson 1991).

References

Hilton, P. and Pederson, J. ``Catalan Numbers, Their Generalization, and Their Uses.'' Math. Intel. 13, 64-75, 1991.